Abelian Varieties


       J.S. Milne




 Version 2.0
 March 16, 2008
These notes are an introduction to the theory of abelian varieties, including the arithmetic
of abelian varieties and Faltings’s proof of certain finiteness theorems. The orginal version
of the notes was distributed during the teaching of an advanced graduate course.




       Alas, the notes are still in very rough form.




BibTeX information
@misc{milneAV,
  author={Milne, James S.},
  title={Abelian Varieties (v2.00)},
  year={2008},
  note={Available at www.jmilne.org/math/},
  pages={166+vi}
}




v1.10 (July 27, 1998). First version on the web, 110 pages.
v2.00 (March 17, 2008). Corrected, revised, and expanded; 172 pages.
Available at www.jmilne.org/math/
Please send comments and corrections to me at the address on my web page.




The photograph shows the Tasman Glacier, New Zealand.




Copyright c 1998, 2008 J.S. Milne.
Single paper copies for noncommercial personal use may be made without explicit permis-
sion from the copyright holder.
Contents

Introduction                                                                                 1

I   Abelian Varieties: Geometry                                                              7
    1    Definitions; Basic Properties. . . . . . . . . . . . . . . . . . . . . . . . . .    7
    2    Abelian Varieties over the Complex Numbers. . . . . . . . . . . . . . . . .        10
    3    Rational Maps Into Abelian Varieties . . . . . . . . . . . . . . . . . . . . .     15
    4    Review of cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . .       20
    5    The Theorem of the Cube. . . . . . . . . . . . . . . . . . . . . . . . . . .       21
    6    Abelian Varieties are Projective . . . . . . . . . . . . . . . . . . . . . . . .   27
    7    Isogenies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    32
    8    The Dual Abelian Variety. . . . . . . . . . . . . . . . . . . . . . . . . . .      34
    9    The Dual Exact Sequence. . . . . . . . . . . . . . . . . . . . . . . . . . .       41
    10 Endomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .        42
    11 Polarizations and Invertible Sheaves . . . . . . . . . . . . . . . . . . . . .       53
    12 The Etale Cohomology of an Abelian Variety . . . . . . . . . . . . . . . .           54
    13 Weil Pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      57
    14 The Rosati Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      61
    15 Geometric Finiteness Theorems . . . . . . . . . . . . . . . . . . . . . . .          63
    16 Families of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . .      67
    17 Néron models; Semistable Reduction . . . . . . . . . . . . . . . . . . . . .        69
    18 Abel and Jacobi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      71

II Abelian Varieties: Arithmetic                                                            75
   1    The Zeta Function of an Abelian Variety . . . . . . . . . . . . . . . . . . .       75
   2    Abelian Varieties over Finite Fields . . . . . . . . . . . . . . . . . . . . . .    78
   3    Abelian varieties with complex multiplication . . . . . . . . . . . . . . . .       83

III Jacobian Varieties                                                                    85
    1    Overview and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
    2    The canonical maps from C to its Jacobian variety . . . . . . . . . . . . . 91
    3    The symmetric powers of a curve . . . . . . . . . . . . . . . . . . . . . . . 94
    4    The construction of the Jacobian variety . . . . . . . . . . . . . . . . . . . 98
    5    The canonical maps from the symmetric powers of C to its Jacobian variety 101
    6    The Jacobian variety as Albanese variety; autoduality . . . . . . . . . . . . 104
    7    Weil’s construction of the Jacobian variety . . . . . . . . . . . . . . . . . . 108
    8    Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
    9    Obtaining coverings of a curve from its Jacobian . . . . . . . . . . . . . . 113


                                              iii
   10    Abelian varieties are quotients of Jacobian varieties . . . . . . . . . . . . . 116
   11    The zeta function of a curve . . . . . . . . . . . . . . . . . . . . . . . . . 117
   12    Torelli’s theorem: statement and applications . . . . . . . . . . . . . . . . 120
   13    Torelli’s theorem: the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 122
   14    Bibliographic notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

IV Finiteness Theorems                                                                     129
   1     Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
   2     The Tate Conjecture; Semisimplicity. . . . . . . . . . . . . . . . . . . . . . 135
   3     Finiteness I implies Finiteness II. . . . . . . . . . . . . . . . . . . . . . . . 139
   4     Finiteness II implies the Shafarevich Conjecture. . . . . . . . . . . . . . . 144
   5     Shafarevich’s Conjecture implies Mordell’s Conjecture. . . . . . . . . . . . 145
   6     The Faltings Height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
   7     The Modular Height. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
   8     The Completion of the Proof of Finiteness I. . . . . . . . . . . . . . . . . . 158

Bibliography                                                                              161

Index                                                                                     165




                                              iv
Notations
We use the standard (Bourbaki) notations: N D f0; 1; 2; : : :g, Z D ring of integers, Q D
field of rational numbers, R D field of real numbers, C D field of complex numbers,
Fp D Z=pZ D field of p elements, p a prime number. Given an equivalence relation, Œ
denotes the equivalence class containing . A family of elements of a set A indexed by a
second set I , denoted .ai /i 2I , is a function i 7! ai W I ! A.
     A field k is said to be separably closed if it has no finite separable extensions of degree
> 1. We use k sep and k al to denote separable and algebraic closures of k respectively. For
a vector space N over a field k, N _ denotes the dual vector space Homk .N; k/.
     All rings will be commutative with 1 unless it is stated otherwise, and homomorphisms
of rings are required to map 1 to 1. A k-algebra is a ring A together with a homomorphism
k ! A. For a ring A, A is the group of units in A:

                  A D fa 2 A j there exists a b 2 A such that ab D 1g:

    df
 X DY      X is defined to be Y , or equals Y by definition;
 X Y      X is a subset of Y (not necessarily proper, i.e., X may equal Y );
 X Y      X and Y are isomorphic;
 X 'Y      X and Y are canonically isomorphic (or there is a given or unique isomorphism).

Conventions concerning algebraic geometry
In an attempt to make the notes as accessible as possible, and in order to emphasize the
geometry over the commutative algebra, I have based them as far as possible on my notes
Algebraic Geometry (AG).
     Experts on schemes need only note the following. An algebraic variety over a field
k is a geometrically reduced separated scheme of finite type over k except that we omit
the nonclosed points from the base space. It need not be connected. Similarly, an algebraic
space over a field k is a scheme of finite type over k, except that again we omit the nonclosed
points.
     In more detail, an affine algebra over a field k is a finitely generated k-algebra R such
that R ˝k k al has no nonzero nilpotents for one (hence every) algebraic closure k al of k.
With such a k-algebra, we associate a ring space Specm.R/ (topological space endowed
with a sheaf of k-algebras), and an affine variety over k is a ringed space isomorphic to
one of this form. AnS   algebraic variety over k is a ringed space .V; OV / admitting a finite
open covering V D Ui such that .Ui ; OV jUi / is an affine variety for each i and which
satisfies the separation axiom. If V is a variety over k and K  k, then V .K/ is the set of
points of V with coordinates in K and VK or V=K is the variety over K obtained from V by
extension of scalars.
     An algebraic space is similar, except that Specm.R/ is an algebraic space for any
finitely generated k-algebra and we drop the separatedness condition.
     We often describe regular maps by their actions on points. Recall that a regular map
W V ! W of k-varieties is determined by the map of points V .k al / ! W .k al / that it
defines. Moreover, to give a regular map V ! W of k-varieties is the same as to give
natural maps V .R/ ! W .R/ for R running over the affine k-algebras (AG 4.37).
     Throughout k is a field.



                                               v
Prerequisites
As a minimum, the reader is assumed to be familiar with basic algebraic geometry, as for
example in my notes AG. Some knowledge of schemes and algebraic number theory will
also be helpful.

References.
In addition to the references listed at the end, I refer to the following of my course notes:
GT Group Theory (v3.00, 2007).
FT Fields and Galois Theory (v4.20, 2008).
AG Algebraic Geometry (v5.10, 2008).
ANT Algebraic Number Theory (v3.00, 2008).
LEC Lectures on Etale Cohomology (v2.01, 1998).
CFT Class Field Theory (v4.00, 2008).

Acknowledgements
I thank the following for providing corrections and comments on earlier versions of these
notes: Holger Deppe, Frans Oort, Bjorn Poonen (and Berkeley students), Vasily Shabat,
Olivier Wittenberg, and others.




                                              vi
Introduction

The easiest way to understand abelian varieties is as higher-dimensional analogues of ellip-
tic curves. Thus we first look at the various definitions of an elliptic curve. Fix a ground
field k which, for simplicity, we take to be algebraically closed. An elliptic curve over k
can be defined, according to taste, as:
  (a) (char.k/ ¤ 2; 3) a projective plane curve over k of the form

                        Y 2 Z D X 3 C aXZ C bZ 3 ;      4a3 C 27b 2 ¤ 0I                 (1)

  (b) a nonsingular projective curve of genus one together with a distinguished point;
  (c) a nonsingular projective curve together with a group structure defined by regular
      maps, or
  (d) (k D C/ an algebraic curve E such that E.C/  C= (as a complex manifold) for
      some lattice  in C.
We briefly sketch the proof of the equivalence of these definitions (see also Milne 2006,
Chapter II).
     (a) !(b). The condition 4a3 C 27b 2 ¤ 0 implies that the curve is nonsingular. Since
it is defined by an equation of degree 3, it has genus 1. Take the distinguished point to be
.0 W 1 W 0/.
     (b) !(a). Let 1 be the distinguished point on the curve E of genus 1. The Riemann-
Roch theorem says that

                        dim L.D/ D deg.D/ C 1          g D deg.D/

where
                        L.D/ D ff 2 k.E/ j div.f / C D  0g:
On taking D D 21 and D D 31 successively, we find that there exists a rational function
x on E with a pole of exact order 2 at 1 and no other poles, and a rational function y on
E with a pole of exact order 3 at 1 and no other poles. The map

                           P 7! .x.P / W y.P / W 1/,   P ¤ 1;
                          1 7! .0 W 1 W 0/

defines an embedding
                                         E ,! P2 :
On applying the Riemann-Roch theorem to 61, we find that there is relation (1) between
x and y, and therefore the image is a curve defined by an equation (1).


                                             1
2                                                                                       INTRODUCTION


     (a,b) !(c): Let Div0 .E/ be the group of divisors of degree zero on E, and let Pic0 .E/
be its quotient by the group of principal divisors; thus Pic0 .E/ is the group of divisor classes
of degree zero on E. The Riemann-Roch theorem shows that the map

                                 P 7! ŒP        Œ1W E.k/ ! Pic0 .E/

is a bijection, from which E.k/ acquires a canonical group structure. It agrees with the
structure defined by chords and tangents, and hence is defined by polynomials, i.e., it is
defined by regular maps.
     (c) !(b): We have to show that the existence of the group structure implies that the
genus is 1. Our first argument applies only in the case k D C. The Lefschetz trace formula
states that for a compact oriented manifold X and a continuous map ˛W X ! X with only
finitely many fixed points, each of multiplicity 1,

            number of fixed points D Tr.˛jH 0 .X; Q//                 Tr.˛jH 1 .X; Q// C    :

If X has a group structure, then, for any nonzero point a 2 X, the translation map ta W x 7!
x C a has no fixed points, and so
                               df
                                   X
                       Tr.ta / D        . 1/i Tr.ta jH i .X; Q// D 0:
                                             i

The map a 7! Tr.ta /W X ! Z is continuous, and so Tr.ta / D 0 also for a D 0. But t0 is
the identity map, and so
                 X
       Tr.t0 / D    . 1/i dim H i .X; Q/ D .X/ (Euler-Poincaré characteristic).

Since the Euler-Poincaré characteristic of a complete nonsingular curve of genus g is 2 2g,
we see that if X has a group structure then g D 1.
     The above argument works over any field when one replaces singular cohomology with
étale cohomology. Alternatively, one can use that if V is an algebraic variety with a group
structure, then the sheaf of differentials is free. For a curve, this means that the canonical
divisor class has degree zero. But this class has degree 2g 2, and so again we see that
g D 1.
     (d) !(b). The Weierstrass }-function and its derivative define an embedding

                                z 7! .}.z/ W } 0 .z/ W 1/ W C= ,! P2 ;

whose image is a nonsingular projective curve of genus 1 (in fact, with equation of the form
(1)).
    (b) !(d). A Riemann surface of genus 1 is of the form C=.

Abelian varieties.
Definition (a) doesn’t generalize — there is no simple description of the equations defining
an abelian variety of dimension1 g > 1. In general, it is not possible to write down explicit
    1 The case g D 2 is something of an exception to this statement. Every abelian variety of dimension 2 is

the Jacobian variety of a curve of genus 2, and every curve of genus 2 has an equation of the form
                                 Y 2 Z 4 D f0 X 6 C f1 X 5 Z C    C f6 Z 6 :
Flynn (1990) has found the equations of the Jacobian variety of such a curve in characteristic ¤ 2; 3; 5 — they
form a set 72 homogeneous equations of degree 2 in 16 variables (they take 6 pages to write out). See Cassels
and Flynn 1996.
                                                                                                               3


equations for an abelian variety of dimension > 1, and if one could, they would be too
complicated to be of use.
    I don’t know whether (b) generalizes. Abelian surfaces are the only minimal surfaces
with the Betti numbers 1; 4; 6; 4; 1 and canonical class linearly equivalent to zero. In general
an abelian variety of dimension g has Betti numbers

                                    1; 2g           2g 
                                          
                                        1 ; : : : ; r ; : : : ; 1:

   Definition (c) does generalize: we can define an abelian variety to be a nonsingular
connected projective2 variety with a group structure defined by regular maps.
   Definition (d) does generalize, but with a caution. If A is an abelian variety over C, then

                                              A.C/  Cg =

for some lattice  in Cg (isomorphism simultaneously of complex manifolds and of groups).
However, when g > 1, the quotient Cg = of Cg by a lattice  does not always arise from
an abelian variety. In fact, in general the transcendence degree over C of the field of mero-
morphic functions Cg = is  g, with equality holding if and only if Cg = is an algebraic
(hence abelian) variety. There is a very pleasant criterion on  for when Cg = is algebraic,
namely, that .Cg ; / admits a Riemann form (see later — Chapter I, 2).

Abelian varieties as generalizations of elliptic curves.
As we noted, if E is an elliptic curve over an algebraically closed field k, then there is a
canonical isomorphism

                                  P 7! ŒP        Œ0W E.k/ ! Pic0 .E/:

    This statement has two generalizations.
    (A) Let C be a curve and choose a point Q 2 C.k/; then there is an abelian variety J ,
called the Jacobian variety of C , canonically attached to C , and a regular map 'W C ! J
such that '.Q/ D 0 and
                       P             P                 0
                          i ni Pi 7!   i ni '.Pi /W Div .C / ! J.k/

induces an isomorphism Pic0 .C / ! J.k/. The dimension of J is the genus of C .
    (B) Let A be an abelian variety. Then there is a “dual abelian variety” A_ such that
Pic .A/ ' A_ .k/ and Pic0 .A_ / ' A.k/ (we shall define Pic0 in this context later). In
    0

the case of an elliptic curve, E _ D E. In general, A and A_ are isogenous, but they are not
equal (and usually not even isomorphic).
    Appropriately interpreted, most of the statements in Silverman’s books on elliptic curves
hold for abelian varieties, but because we don’t have equations, the proofs are more abstract.
In fact, every (reasonable) statement about elliptic curves should have a generalization that
applies to all abelian varieties. However, for some, for example, the Taniyama conjecture,
the correct generalization is difficult to state3 . To pass from a statement about elliptic curves
     2 For historical reasons, we define them to be complete varieties rather than projective varieties, but they

turn out to be projective.
     3 Blasius has pointed out that, by looking at infinity types, one can see that the obvious generalization of

the Taniyama conjecture, namely, that every abelian variety over Q is a quotient of an Albanese variety of a
Shimura variety, can’t be true.
4                                                                                    INTRODUCTION


to one about abelian varieties, replace 1 by g (the dimension of A), and half the copies of
E by A and half by A_ . I give some examples.
    Let E be an elliptic curve over an algebraically closed field k. For any integer n not
divisible by the characteristic, the set of n-torsion points on E, E.k/n , is isomorphic to
.Z=nZ/2 ;and there is a canonical nondegenerate (Weil) pairing

                                      E.k/n  E.k/n ! n .k/

where n .k/ is the group of nth roots of 1 in k. Let A be an abelian variety of dimension g
over an algebraically closed field k. For any integer n not divisible by the characteristic, the
set of n-torsion points on A, A.k/n , is isomorphic to .Z=nZ/2g , and there is a canonical
nondegenerate (Weil) pairing

                                     A.k/n  A_ .k/n ! n .k/.

   Let E be an elliptic curve over a number field k. Then E.k/ is finitely generated
(Mordell-Weil theorem), and there is a canonical height pairing

                                          E.k/  E.k/ ! R

which is nondegenerate module torsion. Let A be an abelian variety over a number field
k. Then A.k/ is finitely generated (Mordell-Weil theorem), and there is a canonical height
pairing
                                   A.k/  A_ .k/ ! R
which is nondegenerate modulo torsion.
   For an elliptic curve E over a number field k, the conjecture of Birch and Swinnerton-
Dyer states that
                                  jTS.E/j jDiscj
                      L.E; s/                 2
                                                  .s 1/r as s ! 1;
                                    jE.k/tors j
where  is a minor term (fudge factor), T S.E/ is the Tate-Shafarevich group of E, Disc is
the discriminant of the height pairing, and r is the rank of E.k/. For an abelian variety A,
Tate generalized the conjecture to the statement
                                         jTS.A/j jDiscj
                       L.A; s/                                .s   1/r as s ! 1:
                                      jA.k/tors j jA_ .k/tors j

We have L.A; s/ D L.A_ ; s/, and Tate proved that jT S.A/j D jT S.A_ /j (in fact the two
groups, if finite, are canonically dual), and so the formula is invariant under the interchange
of A and A_ .4

R EMARK0.1. We noted above that the Betti numbers of an abelian variety    P of dimension  g
are 1; 2g
        1 ;  2g
              2 ; :::; 1. Therefore the Lefschetz trace formula implies that  r .  1/ rC1 2g D
                                                                                           r
0. This can also be proved by using the binomial theorem to expand .1 1/2g .

E XERCISE 0.2. Assume A.k/ and A_ .k/ are finitely generated, of rank r say, and that the
height pairing
                            h; iW A.k/  A_ .k/ ! R
     4 The unscrupulous need read no further: they already know enough to fake a knowledge of abelian vari-

eties.
                                                                                                        5


is nondegenerate modulo torsion. Let e1 ; :::; er be elements of A.k/ that are linearly inde-
pendent over Z, and let f1 ; :::; fr be similar elements of A_ .k/; show that

                                         j det.hei ; fj i/j
                                 .A.k/ W Zei /.A_ .k/ W Zfj /
                                        P                   P

is independent of the choice of the ei and fj . [This is an exercise in linear algebra.]

    The first chapter of these notes covers the basic (geometric) theory of abelian varieties
over arbitrary fields, the second chapter discusses some of the arithmetic of abelian vari-
eties, especially over finite fields, the third chapter is concerned with jacobian varieties, and
the final chapter is an introduction to Faltings’s proof of the Mordell Conjecture.
N OTES . Weil’s books (1948a, 1948b) contain the original account of abelian varieties over fields
other than C, but are written in a language which makes them difficult to read. Mumford’s book
(1970) is the only modern account of the subject, but as an introduction it is rather difficult. It treats
only abelian varieties over algebraically closed fields; in particular, it does not cover the arithmetic
of abelian varieties. Serre’s notes (1989) give an excellent treatment of some of the arithmetic of
abelian varieties (heights, Mordell-Weil theorem, work on Mordell’s conjecture before Faltings —
the original title “Autour du théorème de Mordell-Weil” is more descriptive than the English title.).
Murty’s notes (1993) concentrate on the analytic theory of abelian varieties over C except for the
final 18 pages. The book by Birkenhake and Lange (2004) is a very thorough and complete treatment
of the theory of abelian varieties over C.
Chapter I

Abelian Varieties: Geometry

1    Definitions; Basic Properties.
A group variety over k is an algebraic variety V over k together with regular maps

                            mW V k V ! V          (multiplication)
                                  invW V ! V       (inverse)

and an element e 2 V .k/ such that the structure on V .k al / defined by m and inv is a group
with identity element e.
    Such a quadruple .V; m; inv; e/ is a group in the category of varieties over k. This
means that
                .id;e/             m                      .e;id/         m
           G        ! G k G         ! G;             G       ! G k G     ! G
are both the identity map (so e is the identity element), the maps
                                           id  inv
                                                  !        m
                          G ! G k G                 G k G ! G
                                                   !
                                           inv  id

are both equal to the composite
                                                          e
                                   G ! Specm.k/ ! G

(so inv is the map taking an element to its inverse), and the following diagram commutes
                                                   1m
                              G k G k G             ! G k G
                                   ?                      ?
                                   ?                      ?m
                                   ym1                   y
                                                   m
                                  G k G              !       G
(associativity holds).
    To prove that a group variety satisfies these conditions, recall that the set where two
morphisms of varieties disagree is open (because the target variety is separated, AG 4.8),
and if it is nonempty, then the Nullenstellensatz (AG 2.6) shows that it will have a point
with coordinates in k al .


                                               7
8                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


      It follows that for every k-algebra R, V .R/ acquires a group structure, and these group
structures depend functorially on R (AG 4.42).
      Let V be a group variety over k. For a point a of V with coordinates in k, we define
ta W V ! V (right translation by a) to be the composite
                                                    m
                                 V    ! V V        ! V:
                                 x   7! .x; a/      7! xa

Thus, on points ta is x 7! xa. It is an isomorphism V ! V with inverse tinv.a/ .
     A group variety is automatically nonsingular: it suffices to prove this after k has been
replaced by its algebraic closure (AG, Chapter 11); as does any variety, it contains a non-
singular dense open subvariety U (AG, 5.18), and the translates of U cover V .
     By definition, only one irreducible component of a variety can pass through a nonsin-
gular point of the variety (AG 5.16). Thus a connected group variety is irreducible.
     A connected group variety is geometrically connected, i.e., remains connected when we
extend scalars to the algebraic closure. To see this, we have to show that k is algebraically
closed in k.V / (AG 11.7). Let U be any open affine neighbourhood of e, and let R D
   .U; OV /. Then R is a k-algebra with field of fractions k.V /, and e is a homomorphism
R ! k. If k were not algebraically closed in k.V /, then there would be a field k 0  k,
k 0 ¤ k, contained in R. But for such a field, there is no homomorphism k 0 ! k, which
contradicts the existence of eW R ! k.
     A complete connected group variety is called an abelian variety. As we shall see, they
are projective, and (fortunately) commutative. Their group laws will be written additively.
Thus ta is now denoted x 7! x C a and e is usually denoted 0.

Rigidity
The paucity of maps between projective varieties has some interesting consequences.

T HEOREM 1.1 (R IGIDITY T HEOREM ). Consider a regular map ˛W V  W ! U , and as-
sume that V is complete and that V  W is geometrically irreducible. If there are points
u0 2 U.k/, v0 2 V .k/, and w0 2 W .k/ such that

                           ˛.V  fw0 g/ D fu0 g D ˛.fv0 g  W /

then ˛.V  W / D fu0 g.

   In other words, if the two “coordinate axes” collapse to a point, then this forces the
whole space to collapse to the point.

P ROOF. Since the hypotheses continue to hold after extending scalars from k to k al , we
can assume k is algebraically closed. Note that V is connected, because otherwise V k W
wouldn’t be connected, much less irreducible. We need to use the following facts:
  (i) If V is complete, then the projection map qW V k W ! W is closed (this is the
      definition of being complete AG 7.1).
 (ii) If V is complete and connected, and 'W V ! U is a regular map from V into an affine
      variety, then '.V / D fpointg (AG 7.5). Let U0 be an open affine neighbourhood of
      u0 .
1. DEFINITIONS; BASIC PROPERTIES.                                                        9

                   df
Because of (i), Z D q.˛ 1 .U r U0 // is closed in W . By definition, Z consists of the
second coordinates of points of V  W not mapping into U0 . Thus a point w of W lies
outside Z if and only ˛.V  fwg/  U0 . In particular w0 lies outside Z, and so W r Z
is nonempty. As V  fwg. V / is complete and U0 is affine, ˛.V  fwg/ must be a point
whenever w 2 W r Z: in fact, ˛.V  fwg/ D ˛.v0 ; w/ D fu0 g. Thus ˛ is constant on the
subset V  .W r Z/ of V  W . As V  .W r Z/ is nonempty and open in V  W , and
V  W is irreducible, V  .W r Z/ is dense V  W . As U is separated, ˛ must agree with
the constant map on the whole of V  W .                                              2


C OROLLARY 1.2. Every regular map ˛W A ! B of abelian varieties is the composite of a
homomorphism with a translation.

P ROOF. The regular map ˛ will send the k-rational point 0 of A to a k-rational point b of
B. After composing ˛ with translation by b, we may assume that ˛.0/ D 0. Consider the
map

                         'W A  A ! B,
                          '.a; a0 / D ˛.a C a0 /   ˛.a/   ˛.a0 /:

By this we mean that ' is the difference of the two regular maps
                                              m
                                    AA         ! A
                                     ?            ?
                                     ?˛˛         ?˛
                                     y            y
                                              m
                                   B B         ! B;
which is a regular map. Then '.A  0/ D 0 D '.0  A/ and so ' D 0. This means that ˛
is a homomorphism.                                                                 2


R EMARK 1.3. The corollary shows that the group structure on an abelian variety is uniquely
determined by the choice of a zero element (as in the case of an elliptic curve).

C OROLLARY 1.4. The group law on an abelian variety is commutative.

P ROOF. Commutative groups are distinguished among all groups by the fact that the map
taking an element to its inverse is a homomorphism. Since the negative map, a 7! a,
A ! A, takes the zero element to itself, the preceding corollary shows that it is a homo-
morphism.                                                                              2


C OROLLARY 1.5. Let V and W be complete varieties over k with k-rational points v0 and
w0 , and let p and q be the projection maps V  W ! V and V  W ! W . Let A be an
abelian variety. Then a morphism hW V  W ! A such that h.v0 ; w0 / D 0 can be written
uniquely as h D f ı p C g ı q with f W V ! A and gW W ! A morphisms such that
f .v0 / D 0 and g.w0 / D 0.

P ROOF. Set
                          f D hjV  fw0 g,     g D hjfv0 g  W;
10                                             CHAPTER I. ABELIAN VARIETIES: GEOMETRY


and identify V  fw0 g and fv0 g  W with V and W . On points, f .v/ D h.v; w0 / and
                           df
g.w/ D h.v0 ; w/, and so  D h .f ı p C g ı q/ is the map that sends

                              .v; w/ 7! h.v; w/     h.v; w0 /    h.v0 ; w/:

Thus
                                  .V  fw0 g/ D 0 D .fv0 g  W /
and so the theorem shows that  D 0.                                                       2



2     Abelian Varieties over the Complex Numbers.
Let A be an abelian variety over C, and assume that A is projective (this will be proved in
6). Then A.C/ inherits a complex structure as a submanifold of Pn .C/ (see AG, Chapter
15). It is a complex manifold (because A is nonsingular), compact (because it is closed
in the compact space Pn .C/), connected (because it is for the Zariski topology), and has a
commutative group structure. It turns out that these facts are sufficient to allow us to give
an elementary description of A.C:/

A.C/ is a complex torus.
Let G be a differentiable manifold with a group structure defined by differentiable1 maps
(i.e., a real Lie group). A one-parameter subgroup of G is a differentiable homomorphism
'W R ! G. In elementary differential geometry one proves that for every tangent vector v
to G at e, there is a unique one-parameter subgroup 'v W R ! G such that 'v .0/ D e and
.d'v /.1/ D v (e.g., Boothby 1975, 5.14). Moreover, there is a unique differentiable map

                                           expW Tgte .G/ ! G

such that
                                  t 7! exp.tv/W R ! Tgte .G/ ! G
is 'v for all v; thus exp.v/ D 'v .1/ (ibid. 6.9). When we identify the tangent space at 0 of
Tgte .G/ with itself, then the differential of exp at 0 becomes the identity map

                                          Tgte .G/ ! Tgte .G/:

For example, if G D R , then exp is just the usual exponential map R ! R . If G D
SLn .R/, then exp is given by the usual formula:

                    exp.A/ D I C A C A2 =2Š C A3 =3Š C    , A 2 SLn .R/:

When G is commutative, the exponential map is a homomorphism. These results extend to
complex manifolds, and give the first part of the following proposition.

P ROPOSITION 2.1. Let A be an abelian variety of dimension g over C:
     1 By differentiable I always mean C 1 .
2. ABELIAN VARIETIES OVER THE COMPLEX NUMBERS.                                              11


  (a) There is a unique homomorphism

                                       expW Tgt0 .A.C// ! A.C/

      of complex manifolds such that, for each v 2 Tgt0 .A.C/, z 7! exp.zv/ is the one-
      parameter subgroup 'v W C ! A.C/ corresponding to v. The differential of exp at 0
      is the identity map
                                Tgt0 .A.C// ! Tgt0 .A.C//:
  (b) The map exp is surjective, and its kernel is a full lattice in the complex vector space
      Tgt0 .A.C//:

P ROOF. It remains to prove (b). The image H of exp is a subgroup of A.C/. Because
d.exp/ is an isomorphism on the tangent spaces at 0, the inverse function theorem shows
that exp is a local isomorphism at 0. In particular, its image contains an open neighbourhood
U of 0 in H . But then, for any a 2 H , a C U is an open neighbourhood of a in H , and so
H is open in A.C/. Because the complement of H is a union of translates of H (its cosets),
H is also closed. But A.C/ is connected, and so any nonempty open and closed subset is
the whole space. We have shown that exp is surjective. Denote Tgt0 .A.C// by V , and
regard it as a real vector space of dimension 2g. Recall that a lattice in V is a subgroup of
the form
                                    L D Ze1 C    C Zer
with e1 ; :::; er linearly independent over R; moreover, that a subgroup L of V is a lattice if
and only if it is discrete for the induced topology (ANT 4.14, 4.15), and that it is discrete
if and only if 0 has a neighbourhood U in V such that U \ L D f0g. As we noted above,
exp is a local isomorphism at 0. In particular, there is an open neighbourhood U of 0 such
that exp jU is injective, i.e., such that U \ Ker.exp/ D 0. Therefore Ker.exp/ is a lattice
in V . It must be a full lattice (i.e., r D 2g/ because otherwise V =L  A.C/ wouldn’t be
compact.                                                                                     2


    We have shown that, if A is an abelian variety, then A.C/  Cg =L for some full lattice
L in Cg . However, unlike the one-dimensional case, not every quotient Cg =L arises from
an abelian variety. Before stating a necessary and sufficient condition for a quotient to arise
in this way, we compute the cohomology of a torus.

The cohomology of a torus.
Let X be the smooth manifold V =L where V is real vector space of dimension n and L is
a full lattice in Rn . Note that V D Tgt0 .X/ and L is the kernel of expW V ! X , and so X
and its point 0 determine both V and L. We wish to compute the cohomology groups of X .
    Recall the following statements from algebraic topology (e.g., Greenberg, Lectures on
Algebraic Topology, Benjamin, 1967).

 2.2. (a) Let X be a topological space, and let H  .X; Z/ D r H r .X; Z/; then cup-
                                                                   L
        product defines on H  .X; Z/ a ring structure; moreover

                      ar [ b s D . 1/rs b s [ ar , ar 2 H r .X; Z/, b s 2 H s .X; Z/

      (ibid. 24.8).
12                                        CHAPTER I. ABELIAN VARIETIES: GEOMETRY


  (b) (Künneth formula): Let X and Y be topological spaces such that H r .X; Z/ and
      H s .Y; Z/ are free Z-modules for all r; s. Then there is a canonical isomorphism
                                        M
                      H m .X  Y; Z/ '               H r .X; Z/ ˝ H s .Y; Z/:
                                                  rCsDm

       The map H r .X; Z/ ˝ H s .Y; Z/ ! H rCs .X  Y; Z/ is
                                  a ˝ b 7! p  a [ q  b (cup-product)
      where p and q are the projection maps X  Y ! X; Y .
  (c) If X is a “reasonable” topological space, then
                                    H 1 .X; Z/ ' Hom.1 .X; x/; Z/
      (ibid. 12.1; 23.14).
  (d) If X is compact and orientable of dimension d , the duality theorems (ibid. 26.6,
      23.14) show that there are canonical isomorphisms
                             H r .X; Z/ ' Hd r .X; Z/ ' H d r .X; Z/_
       when all the cohomology groups are torsion-free.

   We first compute the dimension of the groups H r .X; Z/. Note that, as a real manifold,
V =L  .R=Z/n  .S 1 /n where S 1 is the unit circle. We have
                         H r .S 1 ; Z/ D Z; Z; 0; : : : for r D 0; 1; 2; : : : :
Hence, by the Künneth formula,
   H  ..S 1 /2 ; Z/ D Z, Z2 , Z, 0,...
   H  ..S 1 /3 ; Z/ D Z, Z3 , Z3 , Z, 0,...
   H  ..S 1 /4 ; Z/ D Z, Z4 , Z6 , Z4 , Z, 0;.....
The exponents form a Pascal’s triangle:
                                                        !
                                            r     1 n n
                                   dim H ..S / ; Z/ D     :
                                                      r

     Next we compute the groups H r .X; Z/ explicitly. Recall from linear algebraV(e.g.,
                                                                                      r
                   Nr Multilinéaire, Hermann, 1958) that if M is a Z-module, then
Bourbaki, N., Algèbre                                                                  M
is the quotient of    M by the submodule generated by the tensors a1 ˝    ˝ ar in which
two of the ai are equal. Thus,
                      Hom.r M; Z/ ' falternating forms f W M r ! Zg
(a multilinear form is alternating if f .a1 ; :::; ar / D 0 whenever two ai s are equal). If M is
free and finitely generated, with basis e1 ; :::; ed say, over Z, then
                               fe1 ^ : : : ^ eir j i1 < i2 <    < ir g
                Vr
is a basis for   M ; moreover, if M _ is the Z-linear dual Hom.M; Z/ of M , then the
pairing
         ^r       ^r
             M_      M ! Z; .y1 ^ : : : ^ yr ; x1 ^ : : : ^ xr / 7! det.yi .xj //

realizes each of r M _ and r M as the Z-linear dual of the other (ibid. 8, Thm 1).
                V         V
2. ABELIAN VARIETIES OVER THE COMPLEX NUMBERS.                                                  13


T HEOREM 2.3. Let X be the torus V =L. There are canonical isomorphisms
                 ^r                                    ^r
                     H 1 .X; Z/ ! H r .X; Z/ ! Hom.          L; Z/:

P ROOF. For any manifold X , cup-product (2.2a) defines a map
             ^r
                 H 1 .X; Z/ ! H r .X; Z/, a1^ : : :^ ar 7! a1 [ : : : [ ar .

Moreover, the Künneth formula (2.2b) shows that, if this map is an isomorphism for X and
Y and all r, then it is an isomorphism for X  Y and all r. Since this is obviously true for
S 1 , it is true for X  .S 1 /n . This defines the first map and proves that it is an isomorphism.
The space V  Rn is simply connected, and expW V ! X is a covering map — therefore
it realizes V as the universal covering space of X, and so 1 .X; x/ is its group of covering
transformations, which is L. Hence (2.2c)

                                   H 1 .X; Z/ ' Hom.L; Z/:

The pairing
           ^r           ^r
                 L_          L ! Z,    .f1^ : : :^ fr ; e1^ : : :^ er / 7! det .fi .ej //

realizes each group as the Z-linear dual of the other, and L_ D H 1 .X; Z/, and so
                            ^r               '        ^r
                                 H 1 .X; Z/ ! Hom.         L; Z/:                                2




Riemann forms.
By a complex torus, I mean a quotient X D V =L where V is a complex vector space and
L is a full lattice in V .
L EMMA 2.4. Let V be a complex vector space. There is a one-to-one correspondence
between the Hermitian forms H on V and the real-valued skew-symmetric forms E on V
satisfying the identity E.iv; iw/ D E.v; w/, namely,

                               E.v; w/ D Im.H.v; w//I
                               H.v; w/ D E.iv; w/ C iE.v; w/:

P ROOF. Easy exercise.                                                                           2


E XAMPLE 2.5. Consider the torus C=ZCZi . Then

                              E.x C iy; x 0 C iy 0 / D x 0 y     xy 0 ,
                                           H.z; z 0 / D z zN 0

are a pair as in the lemma.

    Let X D V =L be a complex torus of dimension g, and let E be a skew-symmetric form
L  L ! Z. Since L ˝ R D V , we can extend E to a skew-symmetric R-bilinear form
ER W V  V ! R. We call E a Riemann form if
14                                   CHAPTER I. ABELIAN VARIETIES: GEOMETRY


  (a) ER .iv; iw/ D ER .v; w/I
  (b) the associated Hermitian form is positive definite.
    Note that (b) implies that E is nondegenerate, but it is says more.

E XERCISE 2.6. If X has dimension 1, then 2 L  Z, and so there is a skew-symmetric
                                            V
form EW L  L ! Z such that every other such form is an integral multiple of it. The form
E is uniquely determined up to sign, and exactly one of ˙E is a Riemann form.

     We shall say that X is polarizable if it admits a Riemann form.

R EMARK 2.7. Most complex tori are not polarizable. For an example of a 2-dimensional
torus C2 =L with no nonconstant meromorphic functions, see p104 of Siegel 1948.

T HEOREM 2.8. A complex torus X is of the form A.C/ if and only if it is polarizable.

P ROOF ( BRIEF SKETCH ) H)W Choose an embedding A ,! Pn with n minimal. There
exists a hyperplane H in Pn that doesn’t contain the tangent space to any point on A.C/.
Then A \ H is a smooth variety of (complex) dimension g 1 (easy exercise). It can be
“triangulated” by .2g 2/-simplices, and so defines a class in
                                                       2
                                                       ^
                       H2g 2 .A; Z/ ' H 2 .A; Z/ ' Hom. L; Z/;

and hence a skew-symmetric form on L — this can be shown to be a Riemann form.
   (HW Given E, it is possible to construct enough functions (in fact quotients of theta
functions) on V to give an embedding of X into some projective space.                 2


    We define the category of polarizable complex tori as follows: the objects are polariz-
able complex tori; if X D V =L and X 0 D V 0 =L0 are complex tori, then Hom.X; X 0 / is the
set of maps X ! X 0 defined by a C-linear map ˛W V ! V 0 mapping L into L0 . (These are
in fact all the complex-analytic homomorphisms X ! X 0 .)

T HEOREM 2.9. The functor A 7! A.C/ is an equivalence from the category of abelian
varieties over C to the category of polarizable tori.

   In more detail this says that A 7! A.C/ is a functor, every polarizable complex torus is
isomorphic to the torus defined by an abelian variety, and

                            Hom.A; B/ D Hom.A.C/; B.C//:

Thus the category of abelian varieties over C is essentially the same as that of polarizable
complex tori, which can be studied using only (multi-)linear algebra.
    An isogeny of polarizable tori is a surjective homomorphism with finite kernel. The
degree of the isogeny is the order of the kernel. Polarizable tori X and Y are said to be
isogenous if there exists an isogeny X ! Y .

E XERCISE 2.10. Show that “isogeny” is an equivalence relation.
3. RATIONAL MAPS INTO ABELIAN VARIETIES                                                   15


    Let X D V =L. Then

           V  D ff W V ! C j f .˛v/ D N˛f .v/;       f .v C v 0 / D f .v/ C f .v 0 /g

is a complex vector space of the same dimension as V . Define

                              L D ff 2 V  j Im f .L/  Zg:
                                       df
Then L is a lattice in V  , and X _ D V  =L is a polarizable complex torus, called the
dual torus.
E XERCISE 2.11. If X D V =L, then Xm , the subgroup of X of elements killed by m, is
m 1 L=L. Show that there is a canonical pairing

                                  Xm  .X _ /m ! Z=mZ:

This is the Weil pairing.

    A Riemann form on E on X defines a homomorphism E W X ! X _ as follows: let H
be the associated Hermitian form, and let E be the map defined by

                                   v 7! H.v; /W V ! V  :

Then E is an isogeny, and we call such a map E a polarization. The degree of the
polarization is the order of the kernel. The polarization is said to be principal if it is of
degree 1.
E XERCISE 2.12. Show that every polarizable tori is isogenous to a principally polarized
torus.

    A polarizable complex torus is simple if it does not contain a nonzero proper polarizable
subtorus X 0 .
E XERCISE 2.13. Show that every polarizable torus is isogenous to a direct sum of simple
polarizable tori.

    Let E be an elliptic curve over Q. Then End.E/˝Q is either Q or a quadratic imaginary
extension of Q. For a simple polarizable torus, D D End.X/ ˝ Q is a division algebra
over a field and the polarization defines a positive involution  on D. The pairs .D; / that
arise from a simple abelian variety have been classified (A.A. Albert).

Notes
There is a concise treatment of complex abelian varieties in Chapter I of Mumford 1970,
and a more leisurely account in Murty 1993. The classic account is Siegel 1948. Siegel first
develops the theory of complex functions in several variables. See also his books, Topics
in Complex Function Theory. There is a very complete modern account in Birkenhake and
Lange 2004


3    Rational Maps Into Abelian Varieties
Throughout this section, all varieties will be irreducible.
16                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


Rational maps.
We first discuss the general theory of rational maps.
     Let V and W be varieties over k, and consider pairs .U; 'U / where U is a dense open
subset of V and 'U is a regular map U ! W . Two such pairs .U; 'U / and .U 0 ; 'U 0 / are
said to be equivalent if 'U and 'U 0 agree on U \U 0 . An equivalence class of pairs is called
a rational map 'W V _ _ _/ W . A rational map ' is said to be defined at a point v of V if
v 2 U for some .U; 'U / 2 '. The set U1 of v at which ' is definedSis open, and there is
a regular map '1 W U1 ! W such that .U1 ; '1 / 2 ' — clearly, U1 D .U;'U /2' U and we
can define '1 to be the regular map such that '1 jU D 'U for all .U; 'U / 2 '.
     The following examples illustrate the major reasons why a rational map V _ _ _/ W may
fail to extend to a regular map on the whole of V .
  (a) Let W be a proper open subset of V ; then the rational map V _ _ _/ W represented by
      idW W ! W will not extend to V . To obviate this problem, we should take W to be
      complete.
  (b) Let C be the cuspidal plane cubic curve Y 2 D X 3 . The regular map A1 ! C ,
      t 7! .t 2 ; t 3 /, defines an isomorphism A1 r f0g ! C r f0g. The inverse of this
      isomorphism represents a rational map C _ _ _/ A1 which does not extend to a regular
      map on C because the map on function fields doesn’t send the local ring at 0 2 A1
      into the local ring at 0 2 C . Roughly speaking, a regular map can only map a
      singularity to a worse singularity. To obviate this problem, we should take V to be
      nonsingular (in fact, nonsingular is no more helpful than normal).
  (c) Let P be a point on a nonsingular surface V . It is possible to “blow-up” P and
      obtain a surface W and a morphism ˛W W ! V which restricts to an isomorphism
      W r ˛ 1 .P / ! V r P but for which ˛ 1 .P / is the projective line of “directions”
      through P . The inverse of the restriction of ˛ to W r ˛ 1 .P / represents a rational
      map V _ _ _/ W which does not extend to all V , even when V and W are complete —
      roughly speaking, there is no preferred direction through P , and hence no obvious
      choice for the image of P .
     In view of these examples, the next theorem is best possible.

T HEOREM 3.1. A rational map 'W V _ _ _/ W from a normal variety V to a complete variety
W is defined on an open subset U of V whose complement V r U has codimension  2.

P ROOF. Assume first that V is a curve. Thus we are given a nonsingular curve V and a
regular map 'W U ! W from an open subset of V which we want to extend to V . Consider
the maps
                                                  4V
                                              jjjj Ojj
                                            jj jjjjj
                                        jjj
                                                               project
                               '  jjjjj u7!.u;'.u// /
                              U TTTTT                  Z V W
                                        TTTT
                                              TTTT
                                              '   TTTT         project
                                                        TTT* 
                                                            W:
Let U 0 be the image of U in V  W , and let Z be its closure. The image of Z in V
is closed (because W is complete), and contains U (the composite U ! V is the given
inclusion), and so Z maps onto V . The maps U ! U 0 ! U are isomorphisms. Therefore,
3. RATIONAL MAPS INTO ABELIAN VARIETIES                                                     17


Z ! V is a surjective map from a complete curve onto a nonsingular complete curve
that is an isomorphism on open subsets. Such a map must be an isomorphism (complete
nonsingular curves are determined by their function fields). The restriction of the projection
map V  W ! W to Z .' V / is the extension of ' to V we are seeking.
    The general case can be reduced to the one-dimensional case (using schemes). Let U
be the largest subset on which ' is defined, and suppose that V r U has codimension 1.
Then there is a prime divisor Z in V r U . Because V is normal, its associated local ring is
a discrete valuation ring OZ with field of fractions k.V /. The map ' defines a morphism
of schemes Spec.k.V // ! W , which the valuative criterion of properness (Hartshorne
1977, II 4.7) shows extends to a morphism Spec.OZ / ! W . This implies that ' has
a representative defined on an open subset that meets Z in a nonempty set, which is a
contradiction.                                                                               2


Rational maps into abelian varieties.
T HEOREM 3.2. A rational map ˛W V _ _ _/A from a nonsingular variety to an abelian variety
is defined on the whole of V .

P ROOF. Combine Theorem 3.1 with the next lemma.                                             2


L EMMA 3.3. Let 'W V _ _ _/ G be a rational map from a nonsingular variety to a group
variety. Then either ' is defined on all of V or the points where it is not defined form a
closed subset of pure codimension 1 in V (i.e., a finite union of prime divisors).

P ROOF. Define a rational map

                        ˚W V  V _ _ _/ G, .x; y/ 7! '.x/  '.y/ 1 :

More precisely, if .U; 'U / represents ', then ˚ is the rational map represented by
                               'U 'U            id  inv         m
                     U U           ! GG             ! G         ! G:
Clearly ˚ is defined at a diagonal point .x; x/ if ' is defined at x, and then ˚.x; x/ D e.
Conversely, if ˚ is defined at .x; x/, then it is defined on an open neighbourhood of .x; x/;
in particular, there will be an open subset U of V such that ˚ is defined on fxg  U . After
possible replacing U by a smaller open subset (not necessarily containing x/, ' will be
defined on U . For u 2 U , the formula

                                   '.x/ D ˚.x; u/  '.u/

defines ' at x. Thus ' is defined at x if and only if ˚ is defined at .x; x/. The rational map
˚ defines a map
                                  ˚  W OG;e ! k.V  V /:
Since ˚ sends .x; x/ to e if it is defined there, it follows that ˚ is defined at .x; x/ if and
only if
                                  Im.OG;e /  OV V;.x;x/ :
Now V  V is nonsingular, and so we have a good theory of divisors (AG, Chapter 12). For
a nonzero rational function f on V  V , write

                              div.f / D div.f /0     div.f /1 ,
18                                   CHAPTER I. ABELIAN VARIETIES: GEOMETRY


with div.f /0 and div.f /1 effective divisors — note that div.f /1 D div.f      1 / . Then
                                                                                   0

        OV V;.x;x/ D ff 2 k.V  V / j div.f /1 does not contain .x; x/g [ f0g:
Suppose  is not defined at x. Then for some f 2 Im.'  /, .x; x/ 2 div.f /1 , and clearly
˚ is not defined at the points .y; y/ 2  \ div.f /1 . This is a subset of pure codimension
one in  (AG 9.2), and when we identify it with a subset of V , it is a subset of V of
codimension one passing through x on which ' is not defined.                              2


T HEOREM 3.4. Let ˛W V  W ! A be a morphism from a product of nonsingular varieties
into an abelian variety, and assume that V  W is geometrically irreducible. If
                          ˛.V  fw0 g/ D fa0 g D ˛.fv0 g  W /
for some a0 2 A.k/, v0 2 V .k/, w0 2 W .k/, then
                                    ˛.V  W / D fa0 g:

    If V (or W ) is complete, this is a special case of the Rigidity Theorem (Theorem 1.1).
For the general case, we need two lemmas.
L EMMA 3.5. (a) Every nonsingular curve V can be realized as an open subset of a com-
      plete nonsingular curve C .
  (b) Let C be a curve; then there is a nonsingular curve C 0 and a regular map C 0 ! C
      that is an isomorphism over the set of nonsingular points of C .

P ROOF (S KETCH ) (a) Let K D k.V /. Take C to be the set of discrete valuation rings in
K containing k with the topology for which the finite sets and the whole set are closed. For
each open subset U of C , define
                                           \
                                .U; OC / D    fR j R 2 C g:

The ringed space .C; OC / is a nonsingular curve, and the map V ! C sending a point x
of V to OV:x is regular.
    (b) Take C 0 to be the normalization of C .                                     2


L EMMA 3.6. Let V be an irreducible variety over an algebraically closed field, and let P
be a nonsingular point on V . Then the union of the irreducible curves passing through P
and nonsingular at P is dense in V .

P ROOF. By induction, it suffices to show that the union of the irreducible subvarieties of
codimension 1 passing P and nonsingular at P is dense in V . We can assume V to be
affine, and that V is embedded in affine space. For H a hyperplane passing through P but
not containing TgtP .V /, V \ H is nonsingular at P . Let VH be the irreducible component
of V \ H passing through P , regarded as a subvariety of V , and let Z be a closed subset
of V containing all VH . Let CP .Z/ be the tangent cone to Z at P (see AG, Chapter 5).
Clearly,
      TgtP .V / \ H D TgtP .VH / D CP .VH /  CP .Z/  CP .V / D TgtP .V /,
and it follows that CP .Z/ D TgtP .V /. As dim CP .Z/ D dim.Z/ (AG 5.40 et seqq.),
this implies that Z D V (AG 2.26).                                              2
3. RATIONAL MAPS INTO ABELIAN VARIETIES                                                    19


P ROOF ( OF 3.4) Clearly we can assume k to be algebraically closed. Consider first the
case that V has dimension 1. From the (3.5), we know that V can be embedded into a
nonsingular complete curve C , and (3.2) shows that ˛ extends to a map N˛W C  W ! A.
Now the Rigidity Theorem (1.1) shows that ˛N is constant. In the general case, let C be
an irreducible curve on V passing through v0 and nonsingular at v0 , and let C 0 ! C be
the normalization of C . By composition, ˛ defines a morphism C 0  W ! A, which the
preceding argument shows to be constant. Therefore ˛.C  W / D fa0 g, and Lemma 3.6
completes the proof.                                                                  2


C OROLLARY 3.7. Every rational map ˛W G _ _ _/ A from a group variety to an abelian
variety is the composite of a homomorphism hW G ! A with a translation.

P ROOF. Theorem 3.2 shows that ˛ is a regular map. The rest of the proof is the same as
that of Corollary 1.2.                                                                2


Abelian varieties up to birational equivalence.
A rational map 'W V _ _ _/ W is dominating if Im.'U / is dense in W for one (hence all)
representatives .U; 'U / of '. Then ' defines a homomorphism k.W / ! k.V /, and every
such homomorphism arises from a (unique) dominating rational map (exercise!).
    A rational map ' is birational if the corresponding homomorphism k.W / ! k.V / is
an isomorphism. Equivalently, if there exists a rational map W W ! V such that ' ı
and ı ' are both the identity map wherever they are defined. Two varieties V and W are
birationally equivalent if there exists a birational map V _ _ _/ W ; equivalently, if k.V / 
k.W /.
    In general, two varieties can be birationally equivalent without being isomorphic (see
the start of this section for examples). In fact, every variety (even complete and nonsin-
gular) of dimension > 1 will be birationally equivalent to many nonisomorphic varieties.
However, Theorem 3.1 shows that two complete nonsingular curves that are birationally
equivalent will be isomorphic. The same is true of abelian varieties.

T HEOREM 3.8. If two abelian varieties are birationally equivalent, then they are isomor-
phic (as abelian varieties).

P ROOF. Let A and B be the abelian varieties. A rational map 'W A _ _ _/ B extends to a
regular map A ! B (by 3.2). If ' is birational, its inverse also extends to a regular map,
and the composites ' ı and ı ' will be identity maps because they are on open sets.
Hence there is an isomorphism ˛W A ! B of algebraic varieties. After composing it with
a translation, it will map 0 to 0, and then Corollary 1.2 shows that it preserves the group
structure.                                                                                2


P ROPOSITION 3.9. Every rational map A1 _ _ _/ A or P1 _ _ _/ A is constant.

P ROOF. According to (3.2), ˛ extends to a regular map on the whole of A1 . After compos-
ing ˛ with a translation, we may suppose that ˛.0/ D 0. Then ˛ is a homomorphism,

                  ˛.x C y/ D ˛.x/ C ˛.y/;        all x; y 2 A1 .k al / D k al :
20                                       CHAPTER I. ABELIAN VARIETIES: GEOMETRY


But A1      f0g is also a group variety, and similarly,

                   ˛.xy/ D ˛.x/ C ˛.y/ C c;         all x; y 2 A1 .k al / D k al :

This is absurd, unless ˛ is constant.                                                        2


    A variety V over an algebraically closed field is said to be unirational if there is a dom-
inating rational map An _ _ _/ V with n D dim V ; equivalently, if k.V / can be embedded
into k.X1 ; :::; Xn / (pure transcendental extension of k). A variety V over an arbitrary field
k is said to be unirational if Vk al is unirational.
P ROPOSITION 3.10. Every rational map ˛W V _ _ _/ A from a unirational variety to an
abelian variety is constant.

P ROOF. We may suppose that k is algebraically closed. By assumption there is a rational
map An _ _ _/ V with dense image, and the composite of this with ˛ extends to a morphism
ˇW P1      P1 ! A. An induction argument, starting from
                                                           P Corollary 1.5, shows that there
                        1
are regular maps ˇi W P ! A such that ˇ.x1 ; :::; xd / D     ˇi .xi /, and the lemma shows
that each ˇi is constant.                                                                  2



4     Review of cohomology
This section needs to be rewritten (some day, AG will include cohomology).
    In order to prove some of the theorems concerning abelian varieties, we shall need to
make use of results on the cohomology of coherent sheaves. The first of these is Grothendieck’s
relative version of the theorem asserting that the cohomology groups of coherent sheaves
on complete varieties are finite dimensional.
T HEOREM 4.1. If f W V ! T is a proper regular map and F is a coherent OV -module,
then the higher direct image sheaves Rr f F are coherent OT -modules for all r  0.

P ROOF. When f is projective, this is proved in Hartshorne 1977, III 8.8. Chow’s lemma
(ibid. II Ex. 4.10) allows one to extend the result to the general case (EGA III 3.2.1). 2


    The second result describes how the dimensions of the cohomology groups of the mem-
bers of a flat family of coherent sheaves vary.
T HEOREM 4.2. Let f W V ! T be a proper flat regular map, and let F be a locally free
OV -module of finite rank. For each t in T , write Vt for the fibre of V over t and F t for the
inverse image of F on Vt .
    (a) The formation of the higher direct images of F commutes with flat base change. In
        particular, if T D Specm.R/ is affine and R0 is a flat R-algebra, then

                                     H r .V 0 ; F/ D H r .V; F/ ˝R R0

        where V 0 D V Specm R Specm R0 and F 0 is the inverse image of F on V 0 .
    (b) The function
                                      df
                                         X
                         t 7! .F t / D    . 1/r dimk.t / H r .Vt ; F t /
                                                r
        is locally constant on T .
5. THE THEOREM OF THE CUBE.                                                                     21


    (c) For each r, the function
                                       t 7! dimk.t / H r .Vt ; F t /
        is upper semicontinuous (i.e., jumps on closed subsets).
    (d) If T is integral and the function in (c) is constant, then Rr f F is a locally free OT -
        module and the natural maps
                                   Rr f F ˝OT k.t/ ! H r .Vt ; F t /
        are isomorphisms.
    (e) If H 1 .Vt ; F t / D 0 for all t in T , then R1 f F D 0, f F is locally free, and the
        formation of f F commutes with base change.
P ROOF. (a) The statement is local on the base, and so it suffices to prove it for the particular
case in which we have given an explicit statement. In Mumford 1970, 5, p46, a complex
K  of R-modules is constructed such that, for all R-algebras R0 , H r .V 0 ; F 0 / D H r .K  ˝R
R0 /. In our case, R0 is flat over R, and so H r .K  ˝R R0 / D H r .K  / ˝R R0 , which equals
H r .V; F/ ˝R R0 .
     (b), (c), (d). These are proved in Mumford 1970, 5.
     (e) The hypothesis implies that R1 f F D 0 (Hartshorne 1977, III 12.11a), and it fol-
lows that f F ˝OT k.t / ! H 0 .Vt ; F t / is surjective for all t (ibid., III 12.11b) and so is an
isomorphism. Now (ibid., III 12.11b), applied with i D 0, shows that f F is locally free.2


5     The Theorem of the Cube.
We refer the reader to (AG, Chapter 13) for the basic theory of invertible sheaves. For a
variety V , Pic.V / is the group of isomorphism classes of invertible sheaves. [This section
needs to be rewritten to include the complete proof of the theorem of the cube.]

Statement and Applications
Roughly speaking, the theorem of the cube says that an invertible sheaf on the product of
three complete varieties is trivial if it becomes trivial when restricted to each of the three
“coordinate faces”.
T HEOREM 5.1 (T HEOREM OF THE CUBE ). Let U , V , W be complete geometrically irre-
ducible varieties over k, and let u0 2 U.k/, v0 2 V .k/, w0 2 W .k/ be base points. Then
an invertible sheaf L on U  V  W is trivial if its restrictions to
                        U  V  fw0 g, U  fv0 g  W , fu0 g  V  W
are all trivial.

    We defer the proof until later in this section [actually, until the next version].
    Let A be an abelian variety, and let pi W A  A  A ! A be the projection onto the i th
factor (e.g., p2 .x; y; z/ D y), let pij D pi C pj (e.g., p23 .x; y; z/ D y C z), and let
p123 D p1 C p2 C p3 (so that p123 .x; y; z/ D x C y C z).
C OROLLARY 5.2. For any invertible sheaf L on an abelian variety A, the sheaf
                
               p123      
                    L ˝ p12 L 1 ˝ p23
                                   
                                      L 1 ˝ p13
                                             
                                                L 1 ˝ p1 L ˝ p2 L ˝ p3 L
on A  A  A is trivial.
22                                        CHAPTER I. ABELIAN VARIETIES: GEOMETRY


P ROOF. Let m, p, q be the maps A  A ! A sending .x; y/ to x C y, x, y respectively.
The composites of
                       .x; y/ 7! .x; y; 0/W A  A ! A  A  A
with p123 , p12 , p23 ; : : :, p2 ; p3 are respectively m, m, q; : : :, q; 0. Therefore, the restriction
of the sheaf in question to A  A  f0g .' A  A/ is

                 m L ˝ m L 1 ˝ q  L 1 ˝ p  L 1 ˝ p  L ˝ q  L ˝ OAA ;

which is obviously trivial. Similarly, its restriction to f0g  A  A is

                 m L ˝ p  L 1 ˝ m L 1 ˝ q  L 1 ˝ OAA ˝ p  L ˝ q  L,

which is trivial, and its restrictions to A  f0g  A is trivial. Therefore, the theorem of the
cube implies that it is trivial.                                                              2


C OROLLARY 5.3. Let f; g; h be regular maps from a variety V into an abelian variety A.
For any invertible sheaf L on A,

.f C g C h/ L ˝ .f C g/ L 1 ˝ .g C h/ L 1 ˝ .f C h/ L 1 ˝ f  L ˝ g  L ˝ h L

is trivial.

P ROOF. The sheaf in question is the inverse image of the sheaf in (5.2) by the map

                                     .f; g; h/W V ! A  A  A:                                        2


     For an integer n, let nA W A ! A be the map sending an element of A to its nth multiple,
i.e., nA .a/ D a C a C    C a .n summands). This is clearly a regular map; for example,
2A is the composite
                                                        m
                                       A ! A  A ! A:
The map . 1/A sends a to a (it is the map denoted by inv at the start of 1).

C OROLLARY 5.4. For all invertible sheaves L on an abelian variety A,
                                           2                      2
                             nA L  L.n Cn/=2 ˝ . 1/A L.n           n/=2
                                                                             :

In particular,
                                 2
                    nA L  Ln if L is symmetric, i.e., . 1/A L  L:
                    nA L  Ln if L is antisymmetric, i.e., . 1/A L  L 1 :

P ROOF. On applying the last corollary to the maps nA , 1A , . 1/A W A ! A, we find that

                 nA L ˝ .n C 1/A L 1 ˝ .n       1/A L 1 ˝ nA L ˝ L ˝ . 1/A L

is trivial. In other words

                     .n C 1/A L  nA L2 ˝ .n        1/A L 1 ˝ L ˝ . 1/A L                       (2)
5. THE THEOREM OF THE CUBE.                                                                               23


We use this to prove the isomorphism by induction on n. For n D 1, the statement is
obvious. Take n D 1 in (2); then

                          .2/A L  L2 ˝ L ˝ . 1/A L  L3 ˝ . 1/A L

as predicted by the first isomorphism of the lemma. When we assume the Corollary for n,
(2) proves it for n C 1, because

              Œ.n C 1/2 C .n C 1/=2 D .n2 C n/               Œ.n    1/2 C .n      1/=2 C 1
              Œ.n C 1/2       .n C 1/=2 D .n2         n/     Œ.n    1/2     .n   1/=2 C 1:                2



T HEOREM 5.5 (T HEOREM OF THE S QUARE ). For all invertible sheaves L on A and points
a; b 2 A.k/;
                           
                          taCb L ˝ L  ta L ˝ tb L:

P ROOF. On applying (5.3) to the maps x 7! x, x 7! a, x 7! b, A ! A, we find that
                                     
                                    taCb L ˝ ta L 1 ˝ tb L 1 ˝ L

is trivial.                                                                                                 2


R EMARK 5.6. When we tensor the isomorphism in (5.5) with L 2 , we find that
                           
                          taCb L ˝ L 1  .ta L ˝ L 1 / ˝ .tb L ˝ L 1 /:

In other words, the map

                                  a 7! ta L ˝ L 1 W A.k/ ! Pic.A/

is a homorphism. Thus, if a1 C a2 C    C an D 0 (in A.k/), then

                                   ta1 L ˝ ta2 L ˝    ˝ tan L  Ln :

R EMARK 5.7. We can restate the above results in terms of divisors. For a divisor D on A,
write Da for the translate D C a of D. Unfortunately, L.Da / D t a L.D/, but the minus
sign doesn’t matter much because a 7! a is a homomorphism2 (that’s what it means to
be abelian!). Therefore, for any divisor D on A, the map

                                   a 7! ŒDa       DW A.k/ ! Pic.A/

is a homomorphism, where  P Œ denotes the linear equivalence class of . Hence, if a1 C
a2 C    C an D 0, then    Dai  nD.
    For example, let A be an elliptic curve, and let P0 be the zero element of A. Let D0 be
P0 regarded as a divisor of degree 1 on A. For any point P on A, the translate DP of D0
by P is just P regarded as a divisor (i.e., D0 C P D DP /. Therefore, in this case, the last
map is
                             P 7! ŒP P0 W A.k/ ! Pic.A/
as in Milne 2006, 4.10, or Silverman 1986, III 3.4d.
    2 In fact, in this version of the notes, we ignore the sign. Thus, there are some sign differences between

when we express things in terms of divisors and in terms of invertible sheaves.
24                                    CHAPTER I. ABELIAN VARIETIES: GEOMETRY


Preliminaries for the proof of the theorem of the cube.
We list some facts that are required for the proof of the theorem of the cube.

 5.8. Let ˛W M ! N be a homomorphism of free modules of rank 1 over a commutative
ring R. Choose bases e and f for M and N , and set ˛.e/ D rf , r 2 R. If ˛ is surjective,
then r 2 R , and so ˛ is bijective. Consequently, a surjective homomorphism L ! L0 of
invertible sheaves is an isomorphism (because it is on stalks).

 5.9. Let V be a variety over k, and consider the structure map ˛W V ! Specm k. Because
Specm k consists of a single point, to give a coherent sheaf on it the same as to give a finite-
dimensional vector space over k. For a sheaf of OV -modules M on V , ˛ M D .V; M/.
For a vector space M over k, ˛  M D OV ˝k M ; for example, if M D ke1 ˚    ˚ ken ,
then ˛  M D OV e1 ˚    ˚ OV en .

 5.10. Consider a homomorphism R ! S of commutative rings. For any S -module M ,
there is a natural S -linear map

                               S ˝R M ! M , s ˝ m 7! sm:

Similarly, for any regular map ˛W W ! V and coherent OW -module M, there is a canoni-
cal map ˛  ˛ M ! M. For the structure map ˛W V ! Specm k, this is the map

            O V ˝k     .V; M/ ! M, f ˝ m 7! f ˝ .mjU /, f 2              .U; OV /:

 5.11. Consider a homomorphism ˛W M ! N of R-modules. For each maximal ideal m
in R, ˛ induces a homomorphism ˛.m/W M=mM ! N=mN of R=m-vector spaces. If
˛.m/ is surjective, then the homomorphism of Rm -modules ˛m W Mm ! Nm is surjective
(by Nakayama’s lemma).
    Consider a homomorphism ˛W M ! N of coherent OV -modules. For each v 2 V ,
this induces a homomorphism ˛.v/W M.v/ ! N .v/ of k.v/-vector spaces, and if these are
surjective for all v, then Nakayama’s lemma shows that ˛ is surjective. If further M and N
are invertible sheaves, then (5.8) shows that ˛ is an isomorphism.

 5.12. Let V be a complete variety over k, and let M be a locally free sheaf of OV -
modules. For any field K containing k, M defines a sheaf of OVK -modules M0 on VK in
an obvious way, and
                              .VK ; M0 / D .V; M/ ˝k K:
If D is a divisor on a smooth complete variety V , and D 0 is the inverse image of D on VK ,
then
                                 L.D 0 / D L.D/ ˝k K:
Here
                 L.D/ D ff 2 k.V / j div.f / C D  0g D           .V; L.D//
(AG, Chapter 12).

 5.13. Let V be a complete variety, and let L be a locally free sheaf on V . If L becomes
trivial on VK for some field K  k, then it is trivial on V .
5. THE THEOREM OF THE CUBE.                                                                                25


P ROOF. Recall (AG 13.3) that an invertible sheaf on a complete variety is trivial if and only
if it and its dual have nonzero global sections. Thus the statement follows from (4.1).      2


5.14. Consider a regular map V ! T of varieties over k. For any t 2 T , the fibre of the
map over t is a variety over the residue field k.t/:
                                      jt                df
                             V                 Vt     D V k Specm.k.t//
                             #'                # 't
                                      it
                             T                 t      D         Specm.k.t//:

If k is algebraically closed, then k.t/ D k. We can think of the map V ! T as a family of
varieties .Vt / parametrized by the points of T .
     Now let V and T be varieties over k, and consider the projection map qW V  T ! T .
Thus we have the “constant” family of varieties: the fibre Vt D Vk.t / is the variety over
k.t / obtained from V by extending scalars. Let L be an invertible sheaf on V  T . For each
t 2 T , we obtain an invertible sheaf Lt on Vt by pulling back by the map Vt ! V  T .
We regard L as a family of invertible sheaves .Lt / on “V ” parametrized by the points of T .
When k is algebraically closed, Vt D V , and so this is literally true.
     [Add an example.]

 5.15. Let ˛W V ! T be a proper map — for example, ˛ could be the projection map
qW W  T ! T where W is a complete variety (see AG, Chapter 8). For any coherent sheaf
M on V , ˛ M is a coherent sheaf on T .
    Now consider an invertible sheaf L on V  T , and assume that V is complete so that
q L is coherent. The function

                                           t 7! dimk.t /     .Vt ; Lt /

is upper semicontinuous (it jumps on closed subsets). If it is constant, say equal to n,
then q L is locally free of rank n, and the canonical map .q L/.t/ ! .Vt ; Lt / is an
isomorphism.

P ROOF. It is quite difficult to prove that q L is coherent — for a proof in the language of
schemes when V is projective, see Hartshorne 1977, II 5.19.3 Note that, if we assumed that
.q L/.t / had constant dimension then it would follow from (AG 13.1) that q L was locally
free of rank n. However, our assumption that .Vt ; Lt / has constant dimension is easier to
check, and more useful. We omit the proof. See Mumford 1970, II 5.                         2


The seesaw principle.
If an invertible sheaf L on V  T is of the form q  N for some invertible sheaf N on T ,
then Lt is the inverse image of the restriction of N to t, and is therefore trivial. There is a
converse to this statement.
    3 We know that for any complete variety V over k,    .V; OV / D k, which is certainly a finite-dimensional
vector space. When we allow a finite number of poles of bounded order, we still get a finite-dimensional vector
space, i.e., for any divisor D on V , dimk L.D/ is finite. When V is nonsingular, this says that .V; L/ is
finite-dimensional for any invertible sheaf L on V .
26                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


T HEOREM 5.16. Let V and T be varieties over k with V complete, and let L be an invert-
ible sheaf on V  T . If Lt is trivial for all t 2 T , then there exists an invertible sheaf N on
T such that L  q  N .

P ROOF. By assumption, Lt is trivial for all t 2 T , and so .Vt ; Lt /  .Vt ; OV / D k.t/.
                                         df
Therefore (4.2d) shows that the sheaf N D q .L/ is invertible. Consider the canonical map
(5.10)
                                 ˛W q  N D q  q L ! L:
Look at this on the fibre Vt ! Specm k.t/. As Lt  OVt , the restriction of ˛ to Vt
is isomorphic to the natural map (see ??) ˛t W OVt ˝k.t / .Vt ; OVt / ! OVt , which is an
isomorphism. In particular, for any point in w 2 Vt , the map

                                  ˛.w/W .q  N /.w/ ! L.w/

of sheaves on w is an isomorphism. Now (5.11) shows that ˛ is an isomorphism.                  2


C OROLLARY 5.17. Let V and T be varieties over k with V complete, and let L and M be
invertible sheaves on V  T . If Lt  Mt for all t 2 T , then there exists an invertible sheaf
N on T such that L  M ˝ q  N .

P ROOF. Apply (5.16) to L ˝ M 1 .                                                              2


C OROLLARY 5.18 (S EESAW PRINCIPLE ). Suppose that, in addition to the hypotheses of
(5.17), Lv  Mv for at least one v 2 V .k/. Then L  M.

P ROOF. The previous corollary shows that L  M ˝ q  N for some N on T . On pulling
back by the map t 7! .v; t /W T ,! V  T , we obtain an isomorphism Lv  Mv ˝ q  Nv .
As Lv  Mv and .q  N /v D N , this shows that N is trivial.                       2


    The next result shows that the triviality of Lt in the theorem needs only to be checked
for t in some dense subset of T .

P ROPOSITION 5.19. Let V be a complete variety, and let L be an invertible sheaf on V T .
Then ft 2 T j Lt is trivialg is closed in T .

P ROOF. It is the intersection of Supp.q L/ and Supp.q L_ /, which are closed (see AG,
Chapter 13).                                                                           2


Proof of the theorem of the cube.
After (5.12), we may assume that the ground field k is algebraically closed. Because LjU 
V  fw0 g is trivial, the Seesaw Principle and Proposition 5.18 show that it suffices to prove
that Ljz  W is trivial for a dense set of z in U  V . The next lemma shows that we can
assume that V is a curve.

L EMMA 5.20. Let P and Q be points of an irreducible variety over an algebraically closed
field k. Then there is an irreducible curve C on V passing through both P and Q.
6. ABELIAN VARIETIES ARE PROJECTIVE                                                                         27


P ROOF. If V itself is a curve or P D Q, then there is nothing to prove, and so we assume
that dim V > 1 and P ¤ Q: Chow’s lemma (Mumford 1999, p115) says the following:
For any complete variety V , there exists a projective variety W and a surjective birational
morphism W ! V . If we can prove the lemma for W , then clearly we obtain it for V , and
so we may assume V to be projective. By induction on dim V , it suffices to find a proper
closed irreducible subvariety Z of V passing through P and Q. Let 'W V  ! V be the
blow-up of V at fP; Qg. Thus the restriction of ' to V  n' 1 fP; Qg is an isomorphism onto
V n fP; Qg, and the inverse images of P and Q are disjoint divisors on V  . The variety V 
is again projective — we choose a closed immersion V  ,! Pn with n minimal. Bertini’s
Theorem4 states that, for a general hyperplane H in Pn , H \ V  will be irreducible —
here “general” means “for all hyperplanes in an open subset of the dual projective space”.
Choose such an H . Then
                     dim H \ V  C dim ' 1 .P / D 2 dim V                  2  dim V;
and so .H \ V  / \ ' 1 .P / is nonempty (AG 9.23). Similarly, .H \ V  / \ ' 1 .Q/ is
nonempty, and so the image of H \ V  in V is a proper closed irreducible subvariety of V
passing through P and Q.                                                                2


    Thus we can now assume that V is a complete curve, and (by passing to its normal-
ization) a complete nonsingular curve. Now the proof requires nothing more than what we
have proved already and the Riemann-Roch theorem for a curve, and so should have been
included in the notes (Mumford 1970, p57-58). [See the next version.]

Restatement in terms of divisors.
We can restate the above results in terms of divisors. Let V and T be nonsingular varieties
over k with V complete, and let D be a divisor on V  T . There is an open subset of t 2 T
for which, for each prime divisor Z occurring in D, Z \ Vt has codimension one in Vt ,
                                                          df
and, for such t, intersection theory defines a divisor Dt D D  Vt . If Dt  D0 (a constant
divisor on V / for all t in some open subset of T , then
                                        D  D0  T C V  D 0
for some divisor D 0 on T . (This is the original seesaw principle — see Lang 1959, p241).
    Let V and W be complete varieties. A divisorial correspondence between V and W is
a divisor D on V  W . A divisorial correspondence is said to be trivial if it is of the form
V  D C D 0  W where D and D 0 are divisors on V and W . The seesaw principal gives a
criterion for triviality.


6    Abelian Varieties are Projective
We defined an abelian variety to be a complete group variety, and in this section we prove
that it is projective.
    As we saw in the introduction, a projective embedding for an elliptic curve A can be
constructed as follows: let D D P0 where P0 is the zero element of A; for a suitable choice
f1; x; yg of a basis for L.3D/, the map
                                   P 7! .x.P /W y.P /W 1/W A ! P2
    4 Jouanolou, J-P., Théorèmes de Bertini et Applications, Birkhäuser, 1983, 6.3; also Grothendieck’s EGA5.
28                                      CHAPTER I. ABELIAN VARIETIES: GEOMETRY


is an isomorphism of A onto a cubic curve in P2 . We now show how to extend this argument
to any abelian variety.

Embedding varieties in projective space.
For simplicity, in this subsection, we assume k to be algebraically closed; in the next sub-
section, we explain how to remove this condition.
    Let V be a complete nonsingular variety over k. A nonempty linear equivalence class
of effective divisors on V is called a complete linear system. Thus, if d is a complete linear
system and D0 2 d, then d consists of all the effective divisors of the form
                                   D0 C div.f /, f 2 k.V / ;
i.e.,
                              d D fD0 C div.f / j f 2 L.D 0 /g.
For any subspace W  L.D0 /,
                                    fD0 C div.f / j f 2 W g
is called a linear system.
    For example, if V is a closed subvariety of Pn , then
                               fV \ H j H a hyperplane in Pn g
is a linear system. Conversely, we shall associate with a complete linear system on V a
rational map V _ _ _/ Pn , and we shall find conditions on the linear system sufficient to
ensure that the map is an isomorphism of V onto a closed subvariety of Pn :
    Let D0 be a divisor in d, and let f0 ; f1 ; : : : ; fn be a basis for L.D0 /. There is a rational
map
                     P 7! .f0 .P / W f1 .P / W : : : W fn .P //W V _ _ _/ Pn :
It is defined at P provided no fi has a pole at P and at least one fi is nonzero at P — this
is an open set of V .
     When we change the basis, we change the map only by a projective linear transforma-
tion. When we replace D0 by a linearly equivalent divisor, say by D D D0 C div.f /, then
f0 =f; :::; fn =f will be a basis for L.D/, and it defines the same rational map as D. Thus,
up to a projective linear transformation, the rational map depends only on the linear system
d:
     Suppose there exists an effective divisor E such that D  E for all D 2 d. Such an E
                                               df
is called a fixed divisor of d. Clearly, d E D fD E j D 2dg is also a complete linear
system: If D0 2 d, so that d consists of all divisors of the form
                                   D0 C div.f /, f 2 L.D0 /;
then d    E consists of all divisors of the form
                       D0     E C div.f /, f 2 L.D0          E/ D L.D0 /:
Moreover, d E defines the same map into projective space as d:
    Henceforth, we assume that d has no fixed divisor.
    A point P of V is said to be a base point of d if P 2 Supp.D/ for all D 2 d. Every
point of a fixed divisor is a base point but, even when there is no fixed divisor, there may be
base points.
6. ABELIAN VARIETIES ARE PROJECTIVE                                                         29


P ROPOSITION 6.1. The rational map 'W V _ _ _/ Pn defined by d is defined at P if and only
if P is not a base point of d:

P ROOF. Suppose P is not a base point of d, and let D0 be an element of d such that
P … Supp.D0 /. Let f0 ; :::; fn be a basis for L.D0 /. Because d has no fixed divisor,
div.fi =f0 / D Di D0 for some   Di  0. Because P … Supp.D0 /, no fi =f0 can have a
pole at P , and so the map P 7! ff10 .P / W : : : W ffn0 .P / is well-defined at P . 2


    Suppose d has no base points, and let 'W V _ _ _/ Pn be the corresponding rational map.
If ' is an isomorphism onto a closed subvariety of Pn , then

                           d D f' 1 .H / jH a hyperplane in Pn g

(with the grain of salt that ' 1 .H / will not always be a divisor).

D EFINITION 6.2. (a) A linear system d is said to separate points if for any pair of points
      P , Q 2 V , there exists a D 2 d such that

                                 P 2 Supp.D/;       Q … Supp.D/:

  (b) A linear system d is said to separate tangent directions if for every P 2 V and
      nonzero tangent t to V at P , there exists a divisor D 2 d such that P 2 D but
      t … TgtP .D/. (If f is a local equation for D near P , then TgtP .D/ is the subspace
      of TgtP .V / defined by the equation .df /P D 0. Geometrically, the condition means
      that only one prime divisor Z occurring in D can pass through P , that Z occurs with
      multiplicity 1 in D, and that t … TgtP .Z/.)

P ROPOSITION 6.3. Assume that d has no base points. Then the map 'W V ! Pn defined
by d is a closed immersion if and only if d separates points and separates tangent directions.

P ROOF. From the above remarks, the condition is obviously necessary. For the sufficiency,
see, for example, Hartshorne 1977, II 7.8.2.                                            2


T HEOREM 6.4. Every abelian variety A is projective.

P ROOF. The first step is to show that there exists a finite set of prime divisors Zi such that
P
   Zi separates 0 from the remaining points of V , and separates the tangent directions at 0.
More precisely, we want that:
      T
  (a)   Z D f0g (here 0 is the zero element of A);
      T i
  (b)   Tgt0 .Zi / D f0g (here 0 is the zero element of Tgt0 .A/).
To prove this we verify that any two points 0 and P of A are contained in an open affine
subvariety of A. Let U be an open affine neighbourhood of 0, and let U C P be its translate
by P . Choose a point u of U \ .U C P /. Then

                                u 2 U C P H) 0 2 U C P           u;
                          u C P 2 U C P H) P 2 U C P              u;
30                                    CHAPTER I. ABELIAN VARIETIES: GEOMETRY

           df
and so U 0 D U C P u is an open affine neighbourhood of both 0 and P . Identify U 0 with
a closed subset of An , some n. There is a hyperplane H in An passing through 0 but not P ,
and we take Z1 to be the closure of H \ U 0 in A. If there is a P 0 on Z1 other than 0, choose
Z2 to pass through 0 but not P 0 . Continue in this fashion. Because A has the descending
chain
T       condition for closed subsets, this process will end in a finite set of Zi s such that
   Zi D f0g. Now choose any open affine neighbourhood U of P , and let t 2 Tgt0 .P /.
Suppose t 2 Tgt0 .Zi / for all i . Embed U ,! An , and choose a hyperplane H through 0
such that t … H , and add the closure Z of H \ A in A to the set fZi g. Continue in this way
until (b) holds.                  P
    Let D be the divisor Zi where .Zi /1i n satisfies conditions (a) and (b). The sec-
ond step is to show that 3D defines an embedding of A into Pn , some n. For any family
fa1 ; :::; an I b1 ; :::; bn g of points on A, the theorem of the square (5.5, 5.6) shows that
                          X                                     X
                                .Zi;ai C Zi;bi C Zi; ai bi /        3Zi D 3D:
                      i                                      i

This construction gives a very large class of divisors in the complete linear system defined
by 3D. Let a and b be distinct points of A. By (a), for some i , say i D 1, Zi does not
contain b a. Choose a1 D a. Then Z1;a1 passes through a but not b. The sets

                             fb1 j Z1;b1 passes through bg
                             fb1 j Z1; a1 b1 passes through bg

are proper closed subsets of A. Therefore, it is possible to choose a b1 that lies on neither.
Similarly, ai and bi for i  2 can be chosen so   Pthat none of Zi;ai , Zi;bi , or Zi; ai bi
passes through b. Then a is in the support of i .Zi;ai C Zi;bi C Zi; ai bi / but b is
not, which shows that the linear system defined by 3D separates points. The proof that it
separates tangents is similar.                                                              2


Ample divisors.
Let V be a nonsingular complete variety. A divisor D on V is very ample if the complete
linear system it defines gives a closed immersion of V into Pn . A divisor D is ample if nD
is very ample for some n > 0. There are similar definitions for invertible sheaves.
     In the last subsection, we showed that (when k is algebraically closed), there exists an
ample divisor D on an abelian variety A such that 3D is very ample. It is known (but
difficult to prove) that if D is ample on A, then 3D is always very ample.

E XAMPLE 6.5. Let A be an elliptic curve, and let D D 3P0 , where P0 is the zero element
for the group structure. There are three independent functions 1; x; y on A having poles
only at P0 , and there having no worse than a triple pole, that define an embedding of A into
P3 . Thus D is very ample, and P0 (regarded as a divisor) is ample. Since there is nothing
special about P0 (ignoring the group structure), we see that, for any point P , the divisor P
is ample. In fact, it follows easily (from the Riemann-Roch theorem), that D is ample if
and only if deg D > 0, and that if deg D > 3, then D is very ample.
     Something similar is true for any curve C : a divisor D on C is ample if and only if
deg D > 0, and D is very ample if deg D > 2g C 1 (Hartshorne 1977, pp307–308).

   The next proposition removes the condition that k be algebraically closed from Theo-
rem 6.4.
6. ABELIAN VARIETIES ARE PROJECTIVE                                                                           31


P ROPOSITION 6.6. (a) If D and D 0 are ample, so also is D C D 0 .
  (b) If D is an ample divisor on V , then DjW is ample for any closed subvariety W of V
      (assuming DjW is defined).
  (c) A divisor D on V is ample if and only if its extension of scalars to k al is ample on
      Vk al .
  (d) A variety V has an ample divisor if Vk al has an ample divisor.

P ROOF. (a) By definition, there exists an n such that both nD and nD 0 are very ample.
Hence the functions in L.nD/ define an embedding of V into projective space. Because
nD 0 is very ample, it is linearly equivalent to an effective divisor D. Now L.nD C D/ 
L.nD/, and so nD C D is very ample, which implies that nD C nD 0 is very ample (it
defines the same complete linear system as nD C D/.
     (b) The restriction of the map defined by D to W is the map defined by the restriction
of D to W .
     (c) The map obtained by extension of scalars from the map V ! Pn defined by D is
that defined by Dk al (cf. 5.12).
     (d) Let D be an ample divisor on Vk al . Then D will be defined over some finite extension
k 0 of k, and so the set fD j  2 Aut.k al =k/g is finite. Let D0 be the sum of the distinct
D’s — by (a), D0 will be again ample. Then D0 is defined over a finite purely inseparable
extension of k. If k is perfect, then D0 is defined over k; otherwise, p m D0 will be defined
over k for some power p m of the characteristic of k.                                         2

N OTES . We defined an abelian variety to be a complete group variety, and in this section we proved
that it is projective. Of course, we could have avoided this problem by simply defining an abelian
variety to be projective, but this would be historically incorrect.
     In 1940 Weil announced the proof of the Riemann hypothesis for curves over finite fields, based
on a theory of Jacobian varieties of curves over finite fields that did not at the time exist5 . Weil de-
veloped the theory of abelian varieties and Jacobian varieties over fields other than C in the 1940s.
At the time he couldn’t prove that his Jacobian varieties were projective. This forced him to intro-
duce the notion of an “abstract” variety, i.e., a variety that is not embedded in projective space, and
to completely rewrite the foundations of algebraic geometry. In particular, he had to develop a new
intersection theory since the then existing theory used that the variety was embedded in projective
space. In 1946 he published his “Foundations of Algebraic Geometry”, and in 1948 his two books
on abelian varieties and Jacobian varieties in which he proved the Riemann hypothesis for curves
and abelian varieties.
     For me, his work during these years is one of the great achievements of twentieth century math-
ematics, but its repercussions for mathematics were not all good. In his foundations he made little
use of commutative algebra and none of sheaf theory. Beginning in about 1960 Grothendieck com-
pletely rewrote the foundations of algebraic geometry in a way so different from that of Weil that
a generation of mathematicians who had learnt algebraic geometry from Weil’s Foundations found
that they had to learn the subject all over again if they wanted to stay current — many never did.
     About the same time as Weil, Zariski was also rewriting the foundations of algebraic geometry,
but he based his approach on commutative algebra, which leads very naturally into Grothendieck’s
approach. Unfortunately, Zariski did not complete his book on the foundations of algebraic geom-
etry, but only (with the help of Samuel) his volumes on Commutative Algebra (“the child of an
unborn parent”).
     Barsotti (1953), Matsusaka (1953), and Weil (1957) proved that abelian varieties are projective.
Here we presented Weil’s proof.
    5 At the time, April 1940, Weil was in a military prison at Rouen as the result of “un différend avec les

autorités françaises au sujet de mes “obligations” militaires”. Weil said “En d’autres circonstances, une publi-
cation m’aurait paru bien prématurée. Mais, en avril 1940, pouvait-on se croire assuré du lendemain?”
32                                           CHAPTER I. ABELIAN VARIETIES: GEOMETRY


7     Isogenies
Let ˛W A ! B be a homomorphism of abelian varieties. We define the kernel of ˛ to be
the fibre of ˛ over 0 in the sense of algebraic spaces6,7 . It is a closed algebraic subspace
of A, and it is a group in the category of algebraic spaces (a finite group space or scheme).
Hence, if k has characteristic zero, Ker.˛/ is an algebraic variety (AG 11.17d), and hence
equals the fibre over 0 in the sense of algebraic varieties.
    A homomorphism ˛W A ! B of abelian varieties is called an isogeny if it is surjective,
and has finite kernel (i.e., the kernel has dimension zero).

P ROPOSITION 7.1. For a homomorphism ˛W A ! B of abelian varieties, the following are
equivalent:
    (a) ˛ is an isogeny;
    (b) dim A D dim B and ˛ is surjective;
    (c) dim A D dim B and Ker.˛/ is finite;
    (d) ˛ is finite, flat, and surjective.

P ROOF. Because A is complete, ˛.A/ is a closed subvariety of B (AG 7.3c). For any point
b 2 ˛.A/, tb defines an isomorphism of ˛ 1 .0/k.b/ ! ˛ 1 .b/. Thus, up to an extension
of scalars, all fibres of the map ˛ over points of ˛.A/ are isomorphic. In particular, they
have the same dimension. Recall, (AG 10.9) that, for b 2 ˛.A/,

                                dim ˛ 1 .b/  dim A             dim ˛.A/,

and that equality holds on an open set. Therefore the preceding remark shows that, for
b 2 ˛.A/,
                          dim ˛ 1 .b/ D dim A dim ˛.A/:
The equivalence of (a), (b), and (c) follows immediately from this equality. It is clear that
(d) implies (a), and so assume (a). The above arguments show that every fibre has dimension
zero, and so the map is quasi-finite. Now we use the following elementary result: if ˇ ı ˛
is proper and ˇ is separated, then ˛ is proper (Hartshorne 1977, p102). We apply this to the
sequence of maps
                                           ˛
                                        A ! B ! pt
to deduce that ˛ is proper. Now (AG 8.25) shows that ˛, being proper and quasi-finite, is
finite. Hence (see 5.15), ˛ OA is a coherent OB -module, and (AG 13.1) shows that it is
locally free.                                                                          2


     The degree of an isogeny ˛W A ! B is its degree as a regular map, i.e., the degree of
the field extension Œk.A/ W ˛  k.B/. If ˛ has degree d , then ˛ OA is locally free of rank
d . If ˛ is separable, then it is étale (because of the homogeneity, if one point were ramified,
every point would be); if further k is algebraically closed, then every fibre of A ! B has
exactly deg.˛/ points.
     6 In characteristic p, it would cause great confusion to define the kernel to be the fibre in the sense of

algebraic varieties. For example, the formation of the kernel would not commute with extension of the base
field. Unfortunately, the kernel is defined this way in the standard books on Algebraic Groups (but not in my
notes AAG, which include a discussion of this point on p57).
     7 Or schemes if the reader prefers.
7. ISOGENIES                                                                               33


    Recall that nA W A ! A for the regular map that (on points) is

                                   a 7! na D a C    C a:

T HEOREM 7.2. Let A be an abelian variety of dimension g, and let n > 0. Then nA W A !
A is an isogeny of degree n2g . It is always étale when k has characteristic zero, and it is
étale when k has characteristic p ¤ 0 if and only if p does not divide n.

P ROOF. From (6.4, 6.6), we know that there is a very ample invertible sheaf L on A. The
sheaf . 1/A L is again very ample because . 1/A W A ! A is an isomorphism, and so
L ˝ . 1/A L is also ample (see 6.6a). But it is symmetric:

                           . 1/A .L ˝ . 1/A L/ ' L ˝ . 1/A L

because . 1/. 1/ D 1. Thus we have a symmetric very ample sheaf on A, which we
                                                          2
again denote by L. From (5.4) we know that .nA / L  Ln , which is again very ample.
                                           2
Let Z D Ker.nA /. Then .nA / LjZ  Ln jZ, which is both ample and trivial. For a
connected variety V , OV can be very ample only if V consists of a single point. This
proves that Ker.nA / has dimension zero. Fix a very ample symmetric invertible sheaf L,
and write it L D L.D/. Then (AG 12.10),

                       .nA D  : : :  nA D/ D deg.nA /  .D  : : :  D/:

But nA D  n2 D, and so

               .nA D  : : :  nA D/ D .n2 D  : : :  n2 D/ D n2g .D  : : :  D/:

This implies that deg.nA / D n2g , provided we can show that .D  : : :  D/ ¤ 0. But
we chose D to be very ample. Therefore it defines an embedding A ,! Pn , some n, and
the linear system containing D consists of all the hyperplane sections of A (at least, it is
once remove any fixed component). Therefore, in forming .D  : : :  D/ we can replace
D with any hyperplane section of A. We can find hyperplanes H1 ; :::; Hg in Pn such that
H1 \ A; :::; Hg \ A will intersect properly, and then

                        ..H1 \ A/  : : :  .Hg \ A// D deg.A/ ¤ 0:

              T even choose the Hi so that the points of intersection are of multiplicity
(In fact one can
one, so that . Hi / \ A has exactly deg(A) points.) The differential of a homomorphism
˛W A ! B of abelian varieties is a linear map .d˛/0 W Tgt0 .A/ ! Tgt0 .B/. It is true, but
not quite obvious, that
                             d.˛ C ˇ/0 D .d˛/0 C .dˇ/0 ;
i.e., ˛ 7! .d˛/0 is a homomorphism. (The first C uses the group structure on B; the second
uses the vector space structure on Tgt0 .B/; it needs to be checked that they are related.)
Therefore, .d nA /0 D n (multiplication by n, x 7! nx/. Since Tgt0 .A/ is a vector space
over k, this is an isomorphism if char.k/ does not divide n, and it is zero otherwise. In
the first case, nA is étale at 0, and hence (by homogeneity) at every point; in the second it
isn’t.                                                                                      2
34                                       CHAPTER I. ABELIAN VARIETIES: GEOMETRY


R EMARK 7.3. Assume k is separably closed. For any n not divisible by the characteristic
of k,
                                df
                         An .k/ D Ker.nW A.k/ ! A.k//
has order n2g . Since this is also true for any m dividing n, An .k/ must be a free Z=nZ-
module of rank 2g (easy exercise using the structure theorem for finite abelian groups).
    Fix a prime ` ¤ char.k/, and define
                                       T` A D lim A`n .k/:
In down-to-earth terms, an element of T` A is an infinite sequence
                              .a1 ; a2 ; :::; an ; ::::/;   an 2 A.k/;
with `an D an 1 , `a1 D 0 (and so, in particular, an 2 A.k/`n /. One shows that T` A is a
free Z` -module of rank 2g. It is called the Tate module of A.
    When k is not algebraically closed, then one defines
                                    T` A D lim A`n .k sep /:
There is an action of Gal.k sep =k/ on this module, which is of tremendous interest arith-
metically — see later.

R EMARK 7.4. Let k be   ˇ algebraically   closed of characteristic p ¤ 0. In terms of varieties,
all one can say is that ˇAp .k/ˇ D p r , 0  r  g. The typical case is r D g (i.e., this is true
                               ˇ

for the abelian varieties in an open subset of the moduli space). In terms of schemes, one
can show that
                         Ker.pW A ! A/  .Z=pZ/r  ˛p2g 2r  rp ;
where ˛p is the group scheme Spec kŒT =.T p /, and p D Spec kŒT =.T p 1/. Both p
and ˛p are group schemes whose underlying set has a single point. For a k-algebra R,
                                ˛p .R/ D fr 2 R j r p D 0g
                                p .R/ D fr 2 R j r p D 1g:


8    The Dual Abelian Variety.
Let L be an invertible sheaf on A. It follows from the theorem of the square (5.5; 5.6) that
the map
                          L W A.k/ ! Pic.A/, a 7! ta L ˝ L 1
is a homomorphism. Consider the sheaf m L ˝ p  L 1 on A  A, where m and p are the
maps A  A ! A sending .a; b/ to a C b and a respectively. We can regard it as a family
of invertible sheaves on A (first factor) parametrized by A (second factor). Let
                  K.L/ D fa 2 A j .m L ˝ p  L 1 /jA  fag is trivialg:
According to (5.19), this is a closed subset of A. Its definition commutes with extension of
scalars (because of 5.12).
    Note that m ı .x 7! .x; a// D ta and p ı .x 7! .x; a// D id, and so
                         .m L ˝ p  L 1 / j A  fag D ta L ˝ L 1 :
Hence
                            K.L/.k/ D fa 2 A.k/ j L .a/ D 0g:
8. THE DUAL ABELIAN VARIETY.                                                                        35


P ROPOSITION 8.1. Let L be an invertible sheaf such that                 .A; L/ ¤ 0; then L is ample if
and only if K.L/ has dimension zero.

P ROOF. We can suppose that k is algebraically closed (because of 5.12, 6.6). We prove
only that
                     L ample H) K.L/ has dimension zero.
Assume L is ample, and let B be the connected component of K.L/ passing through 0.
                                                   df
It is an abelian variety8 (possible zero) and LB D LjB is ample on B (6.6b). Because,
tb LB  LB for all b 2 B, which implies that the sheaf m LB ˝ p  LB1 ˝ q  LB1 on
B  B is trivial (apply 8.4 below). On taking the inverse image of this sheaf by the regular
map
                                 B ! B  B, b 7! .b; b/
we find that LB ˝. 1B / LB is trivial on B. But, as we saw in the proof of (7.2), LB ample
implies LB ˝ . 1B / LB ample. As in the proof of (7.2), the fact that the trivial invertible
sheaf on B is ample implies that dim B D 0, and so B D 0. [Need to add converse.]          2


R EMARK 8.2. Let D be an effective divisor, and let L D L.D/. By definition, .A; L.D// D
L.D/, and so if D is effective, then .A; L.D// ¤ 0. Therefore, the proposition shows
that that D is ample if and only if the homomorphism

                             D W A.k al / ! Pic.Ak al /, a 7! Da           D;

has finite kernel.

E XAMPLE 8.3. Let A be an elliptic curve, and let D be an effective divisor on A. We have
seen (6.5) that
                           D is ample ” deg.D/ > 0:
Moreover, we know that D D .deg D/2 D0 where D0 D P0 (zero element of A). Hence

                             D has finite kernel ” deg.D/ > 0:

Thus Proposition 8.1 is easy for elliptic curves.

Definition of Pic0 .A/.
For a curve C , Pic0 .C / is defined to be the subgroup of Pic.C / of divisor classes of degree
0. Later, we shall define Pic0 .V / for any complete variety, but first we define Pic0 .A/ for
A an abelian variety. From the formula D D .deg D/2 D in (8.3), on an elliptic curve

                                     deg.D/ D 0 ” D D 0:

This suggests defining Pic0 .A/ to be the set of isomorphism classes of invertible sheaves L
for which L D 0

P ROPOSITION 8.4. For an invertible sheaf on A, the following conditions are equivalent:
  (a) K.L/ D AI
   8 When k is not perfect, it needs to be checked that B is geometrically reduced.
36                                      CHAPTER I. ABELIAN VARIETIES: GEOMETRY


  (b) ta L  L on Ak al , for all a 2 A.k al /I
  (c) m L  p  L ˝ q  L.

P ROOF. The equivalence of (a) and (b) is obvious from the definition of K.L/. Condition
(c) implies that
                     .m L ˝ p  L 1 /j.A  fag/  q  LjA  fag;
which is trivial, and so (c)H)(a). The converse follows easily from the Seesaw Principle
(5.18) because (a) implies that m L ˝ p  L 1 jA  fag and q  LjA  fag are both trivial for
all a 2 A.k al /, and m L ˝ p  L 1 jf0g  A D L D q  Ljf0g  A:                          2


     We define Pic0 .A/ to be the set of isomorphism classes of invertible sheaves satisfying
the conditions of (8.4). I often write L 2 Pic0 .A/ to mean that the isomorphism class of L
lies in Pic0 .A/.

R EMARK 8.5. Let ˛; ˇW V ⇒ A be two regular maps. Their sum ˛ C ˇ is the composite
m ı .˛  ˇ/. If L 2 Pic0 .A/, then

                                  .˛ C ˇ/ L  ˛  L ˝ ˇ  L:

This follows from applying .˛  ˇ/ to the isomorphism in (8.4c). Thus the map

                            Hom.V; A/ ! Hom.Pic0 .A/; Pic.V //

is a homomorphism of groups. In particular,

                                  End.A/ ! End.Pic0 .A//

is a homomorphism. When we apply this to nA D 1A C  C1A , we find that .nA / L  Ln .
                                                 2
Contrast this to the statement that .nA / L  Ln when L is symmetric. They are not
contradictory, because
                            L 2 Pic0 .A/ ) . 1/A L  L 1 ;
i.e., L is antisymmetric.

R EMARK 8.6. Let ˛W A ! B be an isogeny. If Ker.˛/  An , then ˛ factors into
                                    ˛      ˇ
                                 A!B!C             ˇı˛ Dn

and deg.˛/  deg.ˇ/ D n2g . (Because ˛ identifies B with the quotient of A by the subgroup
(scheme) Ker.˛/ (see 8.10), and ˇ exists because of the universal properties of quotients.)

The dual abelian variety.
The points of the dual abelian, or Picard, variety A_ of A should parametrize the elements
of Pic0 .A/.
    Consider a pair .A_ ; P/ where A_ is an algebraic variety over k and P is an invertible
sheaf on A  A_ . Assume
  (a) PjAfbg 2 Pic0 .Ab / for all b 2 A_ , and
  (b) Pjf0gA_ is trivial.
8. THE DUAL ABELIAN VARIETY.                                                               37


We call A_ the dual abelian variety of A, and P the Poincaré sheaf, if the pair .A_ ; P/
has the following universal property: for any pair .T; L/ consisting of a variety T over k
and an invertible sheaf L such that
 (a0 ) LjAft g 2 Pic0 .At / for all t 2 T , and
 (b0 ) Ljf0gT is trivial,
there is a unique regular map ˛W T ! A such that .1  ˛/ P  L.

R EMARK 8.7. (a) If it exists, the pair .A_ ; P/ is uniquely determined by the universal
property up to a unique isomorphism.
    (b) The Picard variety commutes with extension of scalars, i.e., if .A_ ; P/ is the Picard
variety of A over k, then ..A_ /K ; PK / is the Picard variety of AK .
    (c) The universal property says that

         Hom.T; A_ / ' finvertible sheaves on A  T satisfying (a0 ), (b0 )g=  :

In particular
                                      A_ .k/ D Pic0 .A/:
Hence every isomorphism class of invertible sheaves on A lying in Pic0 .A/ is represented
exactly once in the family
                                  fPb j b 2 A_ .k/g:
   (d) The condition (b) is a normalization.
   (e) By using the description of tangent vectors in terms of dual numbers (5.37, one can
show easily that there is a canonical isomorphism

                                  H 1 .A; OA / ! Tgt0 .A_ /:

Cf. the proof of III 2.1 below. In particular, dim A_ D dim A.

L EMMA 8.8. For any invertible sheaf L on A and any a 2 A.k/, ta L ˝ L 1 2 Pic0 .A/.

P ROOF. I prefer to prove this in terms of divisors. Let D be a divisor on A; we have to
show that, for all a 2 A.k/, ŒDa D 2 Pic0 .A/, i.e., that .Da D/b .Da D/  0
for all b 2 A.k/. But

                .Da     D/b     .Da     D/ D DaCb C D       .Da C Db /  0

by the theorem of the square.                                                               2


    Once we’ve shown Picard varieties exist, we’ll see that map A 7! A_ is a functor,
and has the property to be a good duality, namely, A__ ' A. The last statement follows
from the next theorem. First it is useful to define a divisorial correspondence between two
abelian varieties to be an invertible sheaf L on A  B whose restrictions to f0g  B and
A  f0g are both trivial. Let s be the “switch” map .a; b/ 7! .b; a/W A  B ! B  A. If
L is a divisorial correspondence between A and B, then s  L is a divisorial correspondence
between B and A.

T HEOREM 8.9. Let L be a divisorial correspondence between A and B. Then the following
conditions are equivalent:
38                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


  (a) .B; L/ is the dual of A;
  (b) LjA  fbg trivial H) b D 0I
  (c) Ljfag  B trivial H) a D 0I
  (d) .A; s  L/ is the dual of B.

P ROOF. This is not difficult—see Mumford 1970, p81.                                           2


Construction of the dual abelian variety (sketch).
To construct the dual abelian variety, one must form quotients of varieties by the action of
a finite group (group scheme in nonzero characteristic). Since this is quite elementary in
characteristic zero, we sketch the proof. For simplicity, we assume that k is algebraically
closed.

P ROPOSITION 8.10 (E XISTENCE OF QUOTIENTS .). Let V be an algebraic variety over an
algebraically closed field k, and let G be a finite group acting on V by regular maps (on the
right). Assume that every orbit of G is contained in an open affine subset of V . Then there
exists a variety W and a finite regular map W V ! W such that
  (a) as a topological space, .W; / is the quotient of V by G, i.e., W D V =G as a set,
      and U  W is open ”  1 .U / is open;
  (b) for any open affine U  W , .U; OU / D . 1 .U /; OV /G .
The pair .W; / is uniquely determined up to a unique isomorphism by these conditions.
The map  is surjective, and it is étale if G acts freely (i.e., if gx D x H) g D 1).

P ROOF. See Mumford 1970, p66, or Serre 1959, p57.                                             2


     The variety W in the theorem is denoted by V =G and called the quotient of V by G.

R EMARK 8.11. We make some comments on the proof of the proposition.
    (a) It is clear that the conditions determine .W; / uniquely.
    (b) If V is affine, to give an action of G on V on the right is the same as to give an action
of G on .V; OV / on the left. If V D Specm.R/, then clearly we should try defining

                                 W D Specm.S/, S D RG :

To prove (8.10) in this case, one shows that R is a finite R-algebra, and verifies that W has
the required properties. This is all quite elementary.
     (c) Let v 2 V . ByTassumption, there exists an open affine subset U of V containing the
orbit vG of v. Then Ug is again an open affine (see AG 4.27) and contains v; it is also
stable under the action of G. Therefore V is covered by open affines stable under the action
of G, and we can construct the quotient affine by affine, as in (b), and patch them together
to get W .
     (d) The final statement is not surprising: if G acts effectively (i.e., G ! Aut.V / is
injective), then the branch points of the map V ! W are the points x such that Stab.x/ ¤
feg.
     (e) When V is quasi-projective (e.g., affine or projective) every finite set is contained
in an open affine, because for any finite subset of Pn , there exists a hyperplane missing the
8. THE DUAL ABELIAN VARIETY.                                                                 39


set, and we can take U D V \ H (AG 6.25). Therefore each orbit of G is automatically
contained in an open affine subset.
     (f) The pair .W; / has the following universal property: any regular map ˛W V ! W 0
that is constant on the orbits of G in V factors uniquely into ˛ D ˛ 0 ı .
     (g) Lest the reader think that the whole subject of quotients of varieties by finite groups
is trivial, I point out that there exists a nonsingular variety V of dimension 3 on which
G D Z=2Z acts freely and such that V =G does not exist in any reasonable way as an
algebraic variety (Hironaka, Annals 1962). This is a minimal example: the 3 can’t be
replaced by 2, nor the 2 by 1. The quotient fails to exist because there exists an orbit that is
not contained in an open affine subvariety.

R EMARK 8.12. Assume k is algebraically closed. Let A be an abelian variety over k, and
let G be a finite subgroup of A. Then we can form the quotient B D A=G. It is an abelian
variety, and W A ! B is an isogeny with kernel G.
    Recall that an isogeny ˛W A ! B is separable if the field extension k.A/  k.B/ is
separable. This is equivalent to saying that ˛ is étale, because it is then étale at one point
(see AG 10.12b), and so it is étale at all points by homogeneity.
    Let ˛W A ! B be a separable isogeny (for example, any isogeny of degree prime to
the characteristic), and let G D Ker.˛/. From the universal property of A=G, we have a
regular map A=G ! B. This is again separable, and it is bijective. Because B is normal,
this implies that it is an isomorphism (see AG 8.19): B D A=G.
    Now consider two separable isogenies ˇW A ! B, W A ! C , and suppose that
Ker.ˇ/  Ker. /. On identifying B with A= Ker.ˇ/ and using the universal property
of quotients, we find that there is a (unique) regular map ıW B ! C such that ı ı ˇ D .
Moreover, ı is automatically a homomorphism (because it maps 0 to 0).
    For example, suppose ˛W A ! B is a separable isogeny such that Ker.˛/  An . Then
˛ D ˇ ı nA for some isogeny ˇW A ! B, i.e., ˛ is divisible by n in Hom.A; B/.

    Let W D V =G. We shall need to consider the relation between sheaves on V and
sheaves on W . By a coherent G-sheaf on V , we mean a coherent sheaf M of OV -modules
together with an action of G on M compatible with its action on V .

P ROPOSITION 8.13. Assume that the finite group G acts freely on V , and let W D V =G.
The map M 7!   M defines an equivalence from the category of coherent OW -modules
to the category of coherent G-sheaves on V under which locally free sheaves of rank r
correspond to locally free sheaves of rank r.

P ROOF. See Mumford 1970, p70.                                                                2


    The next result is very important.

P ROPOSITION 8.14. If L is ample, then L maps A onto Pic0 .A/.

P ROOF. See Mumford 1970, 8, p77, or Lang 1959, p99.                                         2


    Let L be an invertible sheaf on A, and consider the invertible sheaf

                              L D m L ˝ p  L 1 ˝ q  L 1
40                                         CHAPTER I. ABELIAN VARIETIES: GEOMETRY


on A  A. Then L jf0gA D L ˝ L 1 , which is trivial, and for a in A.k/, L jAfag D
ta L ˝ L 1 D 'L .a/, which, as we have just seen, lies in Pic0 .A/. Therefore, L defines
a family of sheaves on A parametrized by A such that .L /a D 'L .a/. If L is ample,
then (8.14) shows that each element of Pic0 .A/ is represented by .L /a for a (nonzero)
finite number of a in A. Consequently, if .A_ ; P/ exists, then there is a unique isogeny
'W A ! A_ such that .1  '/ P D L . Moreover ' D L , and the fibres of A ! A_ are
the equivalence classes for the relation “a  b if and only if La  Lb ”.
     In characteristic zero, we even know what the kernel of ' is, because it is determined by
its underlying set: it equals K.L/. Therefore, in this case we define A_ to be the quotient
A=K.L/, which exists because of (8.10, 8.11e). The action of K.L/ on the second factor
of A  A lifts to an action on L over A  A, which corresponds by (8.13) to a sheaf P on
A  A_ such that .1  'L / P D L .
     We now check that the pair .A_ ; P/ just constructed has the correct universal property
for families of sheaves M parametrized by normal varieties over k (in particular, this will
imply that it is independent of the choice of L/. Let M on A  T be such a family, and let
                                       
F be the invertible sheaf p12    M ˝ p13  P 1 on A  T  A_ , where pij is the projection
onto the .i; j /th factor. Then

                                      FjA.t;b/  Mt ˝ Pb 1 ;

and so if we let denote the closed subset of T  A_ of points .t; b/ such FjA.t;b/ is
trivial, then   is the graph of a map T ! A_ sending a point t to the unique point b
such that Pb  Ft . Regard        as a closed subvariety of T  A_ . Then the projection
    ! T has separable degree 1 because it induces a bijection on points (see AG 10.12).
As k has characteristic zero, it must in fact have degree 1, and now the original form of
Zariski’s Main Theorem (AG 8.16) shows that ! T is an isomorphism. The morphism
               q
f W T ' ! A_ has the property that .1  f / P D M, as required.
     When k has nonzero characteristic, the theory is the same in outline, but the proofs
become technically much more complicated. The dual variety A_ is still the quotient of
A by a subgroup K.L/ having support K.L/, but K.L/ need not be reduced: it is now
subgroup scheme of V . One defines K.L/ to be the maximal subscheme of A such that the
restriction of m L ˝ q  L 1 to K.L/  A defines a trivial family on A. Then one defines
A_ D A=K.L/. The proof that this has the correct universal property is similar to the
above, but involves much more. However, if one works with schemes, one obtains more,
namely, that .A_ ; P/ has the universal property in its definition for any scheme T . See
Mumford 1970, Chapter III.

R EMARK 8.15. The construction of quotients of algebraic varieties by group schemes is
quite subtle. For algebraic spaces in the sense of Artin, the construction is easier. In partic-
ular, Deligne has proved very general theorem that the quotient of an algebraic space (sense
of Artin) by a finite group scheme exists in a very strong sense.9 Thus, it is more natural to
define A_ as the algebraic space quotient of A by K.L/: The same argument as in 6 then
shows that a complete algebraic space having a group structure is a projective algebraic
variety.
     9 See: David Rydh, Existence of quotients by finite groups and coarse moduli spaces, arXiv:0708.3333,

Theorem 5.4.
9. THE DUAL EXACT SEQUENCE.                                                                   41


9    The Dual Exact Sequence.
Let ˛W A ! B be a homomorphism of abelian varieties, and let PB be the Poincaré sheaf
on B  B _ . According to the definition of the dual abelian variety in the last section, the
invertible sheaf .˛  1/ PB on A  B _ gives rise to a homomorphism ˛ _ W B _ ! A_ such
that .1  ˛ _ / PA  .˛  1/ PB . On points ˛ _ is simply the map Pic0 .B/ ! Pic0 .A/
sending the isomorphism class of an invertible sheaf on B to its inverse image on A.
T HEOREM 9.1. If ˛W A ! B is an isogeny with kernel N , then ˛ _ W B _ ! A_ is an
isogeny with kernel N _ , the Cartier dual of N . In other words, the exact sequence

                                   0!N !A!B!0

gives rise to a dual exact sequence

                                0 ! N _ ! B _ ! A_ ! 0

P ROOF. See Mumford 1970, 15, p143 (case k is algebraically closed), or Oort 1966 (gen-
eral case).                                                                           2


    The statement about the kernels requires explanation. There is a (Cartier) duality theory
N 7! N _ for finite group schemes with the property that N __ ' N . If N has order prime
to the characteristic of k, the duality can be described as follows: when k is separably
closed, N can be identified with the abstract group N.k/, which is finite and commutative,
and
                                 N _ D Hom.N; n .k sep //;
where n is any integer killing N and n is the group of nth roots of 1 in k sep ; when k is not
separably closed, then N.k sep / has an action of Gal.k sep =k/, and N _ .k sep / has the induced
action. When the order of N is not prime to the characteristic, it is more complicated to
describe the duality (see Waterhouse 1979). We mention only that .Z=pZ/_ D p , and
˛p_ D ˛p .
    There is another approach to Theorem 9.1 which offers a different insight. Let L be an
invertible sheaf on A whose class is in Pic0 .A/, and let L be the line bundle associated with
L. The isomorphism p  L˝q  L ! m L of (8.4) gives rise to a map mL W LL ! L lying
over mW A  A ! A. The absence of nonconstant regular functions on A forces numerous
compatibility properties of mL , which are summarized by the following statement.
P ROPOSITION 9.2. Let G.L/ denote L with the zero section removed; then, for some k-
rational point e of G.L/, mL defines on G.L/ the structure of a commutative group variety
with identity element e relative to which G.L/ is an extension of A by Gm :

    Thus L gives rise to an exact sequence

                            E.L/ W 0 ! Gm ! G.L/ ! A ! 0:

    The commutative group schemes over k form an abelian category, and so it is possible
to define Ext1k .A; Gm / to be the group of classes of extensions of A by Gm in this category.
We have:
P ROPOSITION 9.3. The map L 7! E.L/ defines an isomorphism Pic0 .A/ ! Ext1k .A; Gm /:
42                                             CHAPTER I. ABELIAN VARIETIES: GEOMETRY


      Proofs of these results can be found in Serre 1959, VII 3. They show that the sequence
                                    0 ! N _ .k/ ! B _ .k/ ! A_ .k/
can be identified with the sequence of Exts
                       0 ! Homk .N; Gm / ! Ext1k .B; Gm / ! Ext1k .A; Gm /
(The reason for the zero at the left of the second sequence is that Homk .A; Gm / D 0.)
A SIDE 9.4. It is not true that there is a pairing A  A_ !‹, at least not in the category
of abelian varieties. It is possible to embed the category of abelian varieties into another
category which has many of the properties of the category of vector spaces over Q; for
example, it has tensor products, duals, etc. In this new category, there exists a map h.A/ ˝
h.A_ / ! Q which can be thought of as a pairing. The new category is the category of
motives, and h.A/ is the motive attached to A.


10       Endomorphisms
We now write A  B if there exists an isogeny A ! B; then  is an equivalence relation
(because of 8.6).

Decomposing abelian varieties.
An abelian variety A is said to be simple if there does not exist an abelian variety B  A,
0 ¤ B ¤ A.
P ROPOSITION 10.1. For any abelian variety A, there are simple abelian subvarieties A1 ; :::; An 
A such that the map
                                     A1  ::.  An ! A,
                                         .a1 ; :::; an / 7! a1 C    C an
is an isogeny.

P ROOF. By induction, it suffices to prove the following statement: let B be an abelian
subvariety of A, 0 ¤ B ¤ A; then there exists an abelian variety B 0  A such that the map
                                      .b; b 0 / 7! b C b 0 W B  B 0 ! A
is an isogeny. Let i denote the inclusion B ,! A. Choose an ample sheaf L on A, and
define B 0 to be the connected component of the kernel of
                                              i _ ı L W A ! B _
passing through 0. Then B 0 is an abelian variety.10 From (AG 10.9) we know that
                                       dim B 0  dim A            dim B:
The restriction of the morphism A ! B _ to B is LjB W B ! B _ , which has finite kernel
because LjB is ample (8.1, 6.6b). Therefore B \ B 0 is finite, and the map B  B 0 ! A,
.b; b 0 / 7! b C b 0 is an isogeny.                                                   2
     10 This requires proof when k is not perfect, because it is not obvious that B 0 is geometrically reduced (to

be added).
10. ENDOMORPHISMS                                                                            43


R EMARK 10.2. The above proof should be compared with a standard proof (GT 7.5–7.7)
for the semisimplicity of a representation of a finite group G on a finite-dimensional vector
space over Q (say). Let V be a finite dimensional vector space over Q with an action of
G, and let W be a G-stable subspace: we want to construct a complement W 0 to W , i.e., a
G-stable subspace such that the map

                                 w; w 0 7! w C w 0 W W ˚ W 0 ! V;

is an isomorphism. I claim that there is a G-invariant positive-definite form W V  V ! Q.
                                                                 g0 . Let W 0 D W ? . It is
                                                             P
Indeed, let 0 be any positive-definite form, and let  D
stable under G because W is and  is G-invariant. There are at most dim W independent
constraints on a vector to lie in W 0 and so dim W 0  dim V dim W . On the other hand,
jW is nondegenerate (because it is positive-definite), and so W \ W 0 D f0g. This proves
that W ˚ W 0 ' V .
    The form  defines an isomorphism of QŒG-spaces V ! V _ , x 7! .x; /, and W 0 is
the kernel of V ! V _ ! W _ . For abelian varieties, we only have the map A ! A_ , but
in many ways having a polarization on A is like having a positive-definite bilinear form on
A.

    Let A be a simple abelian variety, and let ˛ 2 End.A/. The connected component of
Ker.˛/ containing 0 is an abelian variety,11 and so it is either A or 0. Hence ˛ is either
0 or an isogeny. In the second case, there is an isogeny ˇW A ! A such that ˇ ı ˛ D n,
some n 2 Q. This means that ˛ becomes invertible in End.A/ ˝ Q. From this it follows
that End.A/ ˝ Q is a division algebra, i.e., it is ring, possibly noncommutative, in which
every nonzero element has an inverse. (Division algebras are also called skew fields.) We
let End0 .A/ D End.A/ ˝ Q.
    Let A and B be simple abelian varieties. If A and B are isogenous, then

                              End0 .A/  Hom0 .A; B/  End0 .B/:

More precisely, Hom0 .A; B/ is a vector space over Q which is a free right End0 .A/-
module of rank 1, and a free left End0 .B/ module of rank 1. If they are not isogenous,
then Hom0 .A; B/ D 0.
    Let A be a simple abelian variety, and let D D End0 .A/. Then End.An / ' Mn .D/
(n  n matrices with coefficients in D/.
    Now consider an arbitrary abelian variety A. We have

                                         A  An1 1      Anr r

where each Ai is simple, and Ai is not isogenous to Aj for i ¤ j . The above remarks show
that                               Y
                     End0 .A/         Mni .Di /; Di D End0 .Ai /:

Shortly, we shall see that End0 .A/ is finite-dimensional over Q:
  11 Again, when k is not perfect, it needs to be checked that A is geometrically reduced.
44                                       CHAPTER I. ABELIAN VARIETIES: GEOMETRY


The representation on T` A.
Let A be an abelian variety of dimension g over a field k. Recall that, for any m not divisible
by the characteristic of k, Am .k sep / has order m2g , and that, for a prime ` ¤ char.k/,
T` A D lim A`n .k sep / (see 7.3).

L EMMA 10.3. Let Q be a torsion abelian group, and let (as always) Qn be the subgroup of
elements of order dividing n. Suppose there exists a d such that jQn j D nd for all integers
n. Then Q  .Q=Z/d :

P ROOF. The hypothesis implies that for every n, Qn is a free Z=nZ-module of rank d . The
choice of a basis e1 ; : : : ; ed for Qn determines an isomorphism
                                                          
                        PQn        ! .Z=nZ/d            ! .n 1 Z=Z/d ;
                          ai ei    7! .a1 ; a2 ; : : :/ 7! . an1 ; an2 ; : : :/:

Choose a sequence of positive integers n1 ; n2 ; : : : ; ni ; : : : such each ni divides its successor
ni C1 and every integer divides some ni . Choose a basis e1 ; : : : ; en for Qn1 ; then choose a
basis e10 ; : : : ; en0 for Qn2 such that nn12 ei0 D ei for all i ; and so on. [It must be possible to
say this better!]                                                                                    2


L EMMA 10.4.SLet Q be an `-primary torsion group, and suppose jQ`n j D .`n /d all n > 0.
Set ` 1 Z D ` n Z (inside Q/. Then

                                Q  .` 1 Z=Z/d ' .Q` =Z` /d :

P ROOF. Variant of the above proof.                                                                 2


     These lemmas show that
                                     A.k sep /tors  .Q=Z/2g
(ignoring p-torsion in characteristic p) and that, for ` ¤ char.k/,

                          A.k sep /.`/  .` 1 Z=Z/2g ' .Q` =Z` /2g :

Recall that
                                         Z` D lim Z=`n Z

where the transition maps are the canonical quotient maps Z=`nC1 Z ! Z=`n Z. Thus an
element of Z` can be regarded as an infinite sequence

                                        ˛ D .a1 ; :::; an ; ::::/

with an 2 Z=`n Z and an  an 1 mod `n 1 . Alternatively,

                                        Z` D lim ` n Z=Z

where the transition map ` n 1 Z=Z! ` n Z=Z is multiplication by `. Thus an element of
Z` can be regarded as an infinite sequence

                                        ˛ D .b1 ; :::; bn ; :::/
10. ENDOMORPHISMS                                                                                   45


with bn 2 ` 1 Z=Z, `b1 D 0, and `bn D bn 1 .
    For any abelian group Q, define

                                         T` Q D lim Q`n :

The above discussion shows that T` .` 1 Z=Z/ D Z` . On combining these remarks we
obtain the following result (already mentioned in 6):

P ROPOSITION 10.5. For ` ¤ char.k/, T` A is a free Z` -module of rank 2g.

   A homomorphism ˛W A ! B induces a homomorphism An .k sep / ! Bn .k sep /, and
hence a homomorphism

                                  T` ˛W T` A ! T` B,
                                .a1 ; a2 ; :::/ 7! .˛.a1 /; ˛.a2 /; :::/:

Therefore T` is a functor from abelian varieties to Z` -modules.

L EMMA 10.6. For any prime ` ¤ p, the natural map

                               Hom.A; B/ ! HomZ` .T` A; T` B/

is injective. In particular, Hom.A; B/ is torsion-free.

P ROOF. Let ˛ be a homomorphism such that T` ˛ D 0. Then ˛.P / D 0 for every P 2
A.k sep / such that `n P D 0 for some n. Consider a simple abelian variety A0  A. Then
the kernel of ˛jA0 is not finite because it contains A0`n for all n, and so ˛jA0 D 0. Hence ˛
is zero on every simple abelian subvariety of A, and (10.1) implies it is zero on the whole
of A.                                                                                       2


R EMARK 10.7. Let k D C. The choice of an isomorphism A.C/ ' Cg = determines
isomorphisms An .C/ ' n 1 =. As n 1 = '  ˝ .Z=nZ/,12

                                  T` .A/ ' lim ` n =
                                          ' lim  ˝ .` n Z=Z/
                                          '  ˝ .lim ` n Z=Z/
                                          '  ˝ Z` :

Thus (2.3)
                                     T` A D H1 .A; Z/ ˝ Z` :
One should think of T` A as being “H1 .A; Z` /”. In fact, this is true, not only over C, but
over any field k — T` A is the first étale homology group of A (see LEC).
  12 Tensor products don’t always commute with inverse limits. They do in this case because  is a free

Z-module of finite rank.
46                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


The characteristic polynomial of an endomorphism.
Suppose first that k D C. An endomorphism ˛ of A defines an endomorphism of H1 .A; Q/,
which is a vector space of dimension 2g over Q. Hence the characteristic polynomial P˛
of ˛ is defined:
                            P˛ .X / D det.˛ XjH1 .A; Q//:
It is monic, of degree 2g, and has coefficients in Z (because ˛ preserves the lattice H1 .A; Z/).
More generally, we define the characteristic polynomial of any element of End.A/ ˝ Q by
the same formula.
     We want to define the characteristic polynomial of an endomorphism of an abelian
variety defined over a field an arbitrary field.. Write V` A D T` A ˝Z` Q` .D T` A˝Z Q/.
When k D C, V` A ' H1 .A; Q/ ˝ Q` , and so it natural to try definining

                        P˛ .X / D det.˛      XjV` A/, ` ¤ char.k/:

However, when k ¤ C, it is not obvious that the polynomial one obtains is independent
of `, nor even that it has coefficients in Q. Instead, following Weil, we adopt a different
approach.
L EMMA 10.8. Let ˛ be an endomorphism of a free Z-module  of finite rank such that
˛ ˝ 1W  ˝ Q ! ˝Q is an isomorphism. Then

                                     . W ˛/ D j det.˛/j:

P ROOF. Suppose there exists a basis e1 ; :::; en of  relative to which
                                                                       Q the matrix of ˛ is
diagonal,
Q         say ˛ei D  m i e i for i D 1; : : : ; n.. Then . W ˛/ D  j   mi j and det.˛/ D
   mi . The general case is left as an exercise to the reader. (See Serre 1962, III 1, for
example.)                                                                                2


   Consider an endomorphism ˛ of an abelian variety A over C, and write A D Cg =,
 D H1 .A; Z/. Then Ker.˛/ D ˛ 1 =. If ˛ is an isogeny, then ˛W  !  is injective,
and it defines a bijection

                             Ker.˛/ D ˛ 1 ./ = ! =˛:

Therefore, for an isogeny ˛W A ! A;

                         deg.˛/ D j det.˛jH1 .A; Q//j D jP˛ .0/j:

More generally, for any integer r;

                                     deg.˛   r/ D jP˛ .r/j:

We are almost ready to state our theorem. Let ˛ 2 End.A/. If ˛ is an isogeny, we define
deg.˛/ as before; otherwise, we set deg.˛/ D 0.
T HEOREM 10.9. Let ˛ 2 End.A/. There is a unique monic polynomial P˛ 2 ZŒX of
degree 2g such that P˛ .r/ D deg.˛ r/ for all integers r.

R EMARK 10.10. The uniqueness is obvious: if P and Q are two polynomials such that
P .r/ D Q.r/ for all integers r, then P D Q, because otherwise P   Q would have
infinitely many roots.
10. ENDOMORPHISMS                                                                                    47


R EMARK 10.11. For ˛ 2 End.A/ and n 2 Z;

                        deg.n˛/ D deg.nA /  deg.˛/ D n2g deg.˛/:

We can use this formula to extend the definition of deg to End0 .A/. Since End.A/ is
torsion-free, we can identify End.A/ with a submodule of End0 .A/. For ˛ 2 End0 .A/,
define
                                deg.˛/ D n 2g deg.n˛/
if n is any integer such that n˛ 2 End.A/. The previous formula shows that this is inde-
pendent of the choice of n. Similarly, once we have proved the theorem, we can define

                  P˛ .X / D n 2g Pn˛ .nX/, ˛ 2 End0 .A/, n˛ 2 End.A/:

Then P˛ .X / is a monic polynomial of degree 2g with rational coefficients, and

                               P˛ .r/ D deg.˛       r/, any r 2 Q:

    To prove the theorem we shall prove the following: fix ˛ 2 End0 .A/; then the map
r 7! deg.˛ r/, Q !Q, is given by a polynomial in r of the correct form. In fact, we shall
prove a little more.
    A function f W V ! K on a vector space V over a field K is said to be a polynomial
function of degree d if for every finite linearly independent set fe1 ; :::; en g of elements of
V , f .x1 e1 C    C xn en / is a polynomial function of degree d in the xi with coefficients
in K (i.e., there is a polynomial P 2 KŒX1 ; : : : ; Xn  such that f .x1 e1 C    C xn en / D
P .x1 ; : : : ; xn / for all .x1 ; : : : ; xn / 2 K n ). A homogeneous polynomial function is defined
similarly.

L EMMA 10.12. Let V be a vector space over an infinite field K, and let f W V ! K be a
function such that, for all v; w in V , x 7! f .xv C w/W K ! K is a polynomial in x with
coefficients in K; then f is a polynomial function.

P ROOF. We show by induction on n that, for every subset fv1 ; :::; vn ; wg of V , f .x1 v1 C
   C xn vn C w/ is a polynomial in the xi . For n D 1, this is true by hypothesis; assume it
for n 1. The original hypothesis applied with v D vn shows that

        f .x1 v1 C    C xn vn C w/ D a0 .x1 ; ::; xn 1 / C    C ad .x1 ; :::; xn 1 /xnd

for some d , with the ai functions k n 1 ! k. Choose distinct elements c0 ; :::; cd of K; on
solving the system of linear equations

   f .x1 v1 C    C xn 1 vn 1 C cj vn C w/ D ˙ ai .x1 ; :::; xn 1 /cji ;       j D 0; 1; :::; d;

for ai , we obtain an expression for ai as a linear combination of the terms f .x1 v1 C
   C xn 1 vn 1 C cj vn C w/, which the induction assumption says are polynomials in
x1 ; :::; xn 1 .                                                                      2


P ROPOSITION 10.13. The function ˛ 7! deg.˛/W End0 .A/!Q is a homogeneous poly-
nomial function of degree 2g in End0 .A/:
48                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


P ROOF. According to the lemma, to show that deg.˛/ is a polynomial function, it suffices
to show that deg.n˛ C ˇ/ is a polynomial in n for fixed ˛; ˇ 2 End0 .A/. But we already
know that deg is homogeneous of degree 2g, i.e., we know

                                  deg.n˛/ D n2g deg.˛/;

and using this one sees that it suffices to prove that deg.n˛ C ˇ/ is a polynomial of degree
 2g for n 2 Z and fixed ˛; ˇ 2 End.A/. Let D be a very ample divisor on A, and let
Dn D .n˛ C ˇ/ D. Then (AG 12.10)

                       .Dn  : : :  Dn / D deg.n˛ C ˇ/  .D  : : :  D/

and so it suffices to show that .Dng / is a polynomial of degree  2g in n. Corollary (5.3)
applied to the maps n˛ C ˇ, ˛, ˛W A ! A and the sheaf L D L.D/ shows that

                      DnC2    2DnC1        .2˛/ D C Dn C 2.˛  D/  0

i.e.,
             DnC2      2DnC1 C Dn D D 0 , where D 0 D .2˛/ D               2.˛  D/:
An induction argument now shows that

                                 n.n       1/
                         Dn D                   D 0 C nD1    .n     1/D0
                                       2
and so
                                                       n.n        1/ g 0g
                deg.n˛ C ˇ/  .D g / D .Dng / D .                   / .D / C : : :
                                                             2
which is a polynomial in n of degree  2g.                                                2


P ROOF ( OF T HEOREM 10.9) Proposition 10.13 shows that, for each ˛ in End0 .A/, there
is a polynomial P˛ .X / 2 QŒX  of degree 2g such that, for all rational numbers r, P˛ .r/ D
deg.˛ rA /. It remains to show that P˛ is monic and has integer coefficients when ˛ 2
End.A/. Let D be an ample symmetric divisor on A; then
                         df
               P˛ . n/ D deg.˛ C n/ D .Dng /=.D g /, Dn D .˛ C n/ D;

and the calculation in the proof of (10.13) shows that

                      Dn D .n.n      1/=2/D 0 C .˛ C nA / D C ˛  D;

with D 0 D .2A / D     2D  2D. It follows now that P˛ is monic and that it has integer
coefficients.                                                                             2


    We call P˛ the characteristic polynomial of ˛ and we define the trace Tr.˛/ of ˛ by
the equation
                    P˛ .X / D X 2g Tr.˛/X 2g 1 C    C deg.˛/:
10. ENDOMORPHISMS                                                                              49


The representation on T` A (continued).
We know that Hom.A; B/ injects into HomZ` .T` A; T` B/, which is a free Z` -module of
rank 2 dim.A/  2 dim.B/. Unfortunately, this doesn’t show that Hom.A; B/ is of finite
rank, because Z` is not finitely generated as a Z-module. What we need is that
      e1 ; :::; er linearly independent over Z H) T` .e1 /; :::; T` .er / linearly independent over Z` ;
or equivalently, that
                              Hom.A; B/ ˝ Z` ! Hom.T` A; T` B/
is injective.

A SIDE 10.14. The situation is similar to that in which we have a Z-module M contained
in a finite-dimensional real vector space V . In that case we want M to be a lattice in V .
Clearly, M needn’t be finitely generated, but even if it is, it needn’t be a lattice — consider
                                         p
                            M D fm C n 2 j m; n 2 Zg  R:

The way we usually prove that such an M is a lattice is to prove that it is discrete in V .
Here we use the existence of P˛ to prove something similar for Hom.A; B/.

T HEOREM 10.15. For any abelian varieties A and B, and ` ¤ char.k/, the natural map

                           Hom.A; B/ ˝ Z` ! Hom.T` A; T` B/

is injective, with torsion-free cokernel. Hence Hom.A; B/ is a free Z-module of finite rank
 4 dim.A/ dim.B/.

L EMMA 10.16. Let ˛ 2 Hom.A; B/; if ˛ is divisible by `n in Hom.T` A; T` B/, then it is
divisible by `n in Hom.A; B/.

P ROOF. The hypothesis implies that ˛ is zero on A`n , and so we can apply the last state-
ment in (8.12) to write ˛ D ˇ ı `n .                                                    2


P ROOF ( OF T HEOREM 10.15) We first prove (10.15) under the assumption that Hom.A; B/
Pfinitely generated over Z. Let e1 ; :::; em be a basis for Hom.A; B/, and suppose that
is
    ai T` .ei / D 0 with ai 2 Z` . For each i , choose a sequence of integers ni .r/ converging
`-adically to ai . Then jni .r/j` is constant for r large, i.e.,
                                                            P the power ofP` dividing ni .r/
doesn’t change after a certain point. But for r large T` .       ni .r/ei / D     ni .r/T` .ei / is
close to zero in Hom.T` A; T` B/, which means that it is divisible by a high power of `, and
so each ni .r/ is divisible by a high power of `. The contradicts the earlier statement.
    Thus it remains to prove that Hom.A; B/ is finitely generated over Z. We first show
that End.A/ is finitely generated when A is simple. Let e1 ; :::; em be linearly independent
over Z in End.A/. Let P be the polynomial function on End0 .A/ such that P .˛/ D
deg.˛/ for all ˛ 2 End.A/. Because A is simple, a nonzero endomorphism ˛ of A is
an isogeny, and so P .˛/ is an integer > 0. Let M be the Z-submodule of End.T` A/
generated by the ei . The map P W QM ! Q is continuous for the real topology because it
is a polynomial in the coordinates, and so U D fvjP .v/ < 1g is an open neighbourhood
of 0. As .QM \ End A/ \ U D 0, we see that QM \ End.A/ is discrete in QM , and
therefore is a finitely generated Z-module (ANT 4.15). Now choose the ei to be a Q-basis
for End0 .A/. Then QM \ End.A/ D End.A/, which is therefore finitely generated.
50                                    CHAPTER I. ABELIAN VARIETIES: GEOMETRY

                                                             Q s
   For arbitrary A; B choose isogenies i Ari i ! A and B ! j Bj j with the Ai and
                                      Q
Bj simple. Then                         Y
                          Hom.A; B/ !          Hom.Ai ; Bj /
                                                 i;j

is injective. As Hom.Ai ; Bj / D 0 if Ai and Bj are not isogenous, and Hom.Ai ; Bj / ,!
End.Ai / if there exists an isogeny Bj ! Ai , this completes the proof.               2


R EMARK 10.17. Recall that for a field k, the prime field of k is its smallest subfield. Thus
the prime field of k is Q if char.k/ D 0 and it is Fp if char.k/ D p ¤ 0. Suppose that k
is finitely generated over its prime field k0 , so that k has finite transcendence degree and is
a finite extension of a pure transcendental extension. For example, k could be any number
field or any finite field. Let D Gal.k al =k/, and let A and B be abelian varieties over k.
In 1964, Tate conjectured that

                          Hom.A; B/ ˝ Z` ! Hom.T` A; T` B/

is an isomorphism. Here the superscript means that we take only the Z` -linear homo-
morphisms T` A ! T` B that commute with the action of .
    Tate proved this in 1966 for a finite field; Zarhin proved it for many function fields in
characteristic p, and Faltings proved in characteristic 0 in the same paper in which he first
proved the Mordell conjecture — see Chapter IV, 2, below.

The Néron-Severi group.
For a complete nonsingular variety V , Pic.V /= Pic0 .V / is called the Néron-Severi groupNS.V /
of V . Severi proved that NS.V / is finitely generated for varieties over C, and Néron proved
the same result over any field k (whence the name). Note that, for a curve C over an al-
gebraically closed field k, the degree map gives an isomorphism NS.C / ! Z (if k is not
algebraically closed, the image may be of finite index in Z, i.e., the curve may not have a
divisor class of degree 1).
    For abelian varieties, we can prove something stronger than the Néron-Severi theorem.
C OROLLARY 10.18. The Néron-Severi group of an abelian variety is a free Z-module of
rank  4  dim.A/2 .

P ROOF. Clearly L 7! L defines an injection NS.A/ ,! Hom.A; A_ /, and so this follows
from (10.15).                                                                        2


R EMARK 10.19. The group NS.A/ is a functor of A. Direct calculations show that ta acts
as the identity on NS.A/ for all a in A.k/ (because ta L D L / and n acts as n2 (because
                                2
  1 acts as 1, and so n L  Ln in NS.A/ by 5.4).

The representation on T` A (continued).
As we noted above, P˛ should be the characteristic polynomial of ˛ acting on V` A for any
` ¤ char.k/. Here we verify this.
P ROPOSITION 10.20. For all ` ¤ char.k/, P˛ .X/ is the characteristic polynomial of ˛
acting on V` A; hence the trace and degree of ˛ are the trace and determinant of ˛ acting on
V` A.
10. ENDOMORPHISMS                                                                              51


    We need two elementary lemmas.
                              Q                           Q
L EMMA 10.21. Let P .X / D .X ai / andQQ.X/ D .XQbi / be monic polynomials of
the same degree with coefficients in Q` ; if j i F .ai /j` D j i F .bi /j` for all F 2 ZŒT ,
then P D Q.

P ROOF. By continuity, P and Q will satisfy the condition for all F with coefficients in Z` ,
and even in Q` . Let d and e be the multiplicities of a1 as a root of P and Q respectively
— we shall prove the lemma by verifying that d D e. Let ˛ 2 Qal    `
                                                                      be close to a1 , but not
equal to a1 . Then

                          jP .˛/j` D j˛ a1 jd` ai ¤a1 j˛ ai j
                                                Q

                         jQ.˛/j` D j˛ a1 je` bi ¤a1 j˛ bi j:
                                                Q

Let F be the minimum polynomial of ˛ over Q` , and let m D deg F . Let ˙ be a set of
automorphisms  of Qal
                    `
                       such f ˛ j  2 ˙g is the set of distinct conjugates of ˛. Then,
                             Q             Q
                               i F .ai / D   ;i .ai  ˛/:

Because  permutes the ai ,
                              Q                         Q
                                   i .ai    ˛/ D           i .ai        ˛/;

and because the automorphisms of Qal
                                  `
                                     preserve valuations,

                                   j. ai     ˛/j` D jai            ˛j` :

Hence
                                                                     ˛/jm
                                  Q                  Q
                              j       i F .ai /j` D j    i .ai          ` :

Similarly,
                                                                     ˛/jm
                                   Q                    Q
                               j       i F .bi /j` D j      i .bi       `

and so our hypothesis implies that

                 j˛ a1 jd` ai ¤a1 j˛                             a1 je`
                          Q                                               Q
                                               ai j D j˛                   bi ¤a1 j˛   bi j:

As ˛ approaches a1 the factors not involving a1 will remain constant, from which it follows
that d D e.                                                                               2


L EMMA 10.22. Let E be an algebra over a field K, and let ıW E ! K be a polynomial
                                     Q over K/ such that ı.˛ˇ/ D ı.˛/ı.ˇ/ for all
function on E (regarded as a vector space
˛; ˇ 2 E. Let ˛ 2 E, and letQ P D i .X ai / be the polynomial such that P .x/ D
ı.˛ x/. Then ı.F .˛// D ˙ i F .ai / for any F 2 KŒT .

P ROOF. After extending K, we may assume that the roots b1 ; b2 ; :.. of F and of P lie in
K; then
                     Q             Q            Q                        Q
        ı.F .˛// D ı. j .˛ bj // D j P .bj / D i;j .bj ai / D ˙ i F .ai /:               2
52                                            CHAPTER I. ABELIAN VARIETIES: GEOMETRY


P ROOF. We now prove (10.20). Clearly we may assume k D ks . For any ˇ 2 End.A/,
                                     j deg.ˇ/j` D j#.Ker.ˇ//j`
                                                  D #.Ker.ˇ/.`// 1
                                                  D #.Coker.T` ˇ// 1
                                                  D j det.T` ı ˇ/j` :
Consider ˛ 2 End.A/, and let a1 ; a2 ; :.. be the roots of P˛ . Then for any polynomial
F 2 ZŒT , by 10.22,
              Y                                                     Y
            j    F .ai /j` D j deg F .˛/j` D j det T` .F .˛//j` D j   F .bi /j`

where the bi are the eigenvalues of T` ˇ. By Lemma 10.21, this proves the proposition.                         2


Study of the endomorphism algebra of an abelian variety
Let D be a simple algebra (not necessarily commutative) of finite-degree over its centre K
(a field). The reduced trace and reduced norm of D over K satisfy
                                                           1
                                TrD=K .˛/ D ŒD W K 2 TrdD=K .˛/,
                                                                       1
                              NmD=K .˛/ D NrdD=K .˛/ŒDWK , ˛ 2 D:
                                                                       2



When D is a matrix algebra Mr .K/, then the reduced trace of ˛ is the trace of ˛ regarded
as a matrix and the reduced norm of ˛ is its determinant. In general, D ˝K L  Mr .L/ for
some finite Galois extension L of K, and the reduced trace and norm of an element of D
can be defined to be the trace and determinant of its image in Mr .L/ — these are invariant
under the Galois group, and so lie in K. Similarly, the reduced characteristic polynomial of
˛ in D=K satisfies
                            PD=K;˛ .X/ D .PrdD=K;˛ .X//ŒDWK :
      For a simple algebra D of finite degree over Q, we define
                  TrD=Q D TrK=Q ı Tr dD=K ;              NmD=Q D NmK=Q ıNrdD=K ;
where K is the centre of D. Similarly, we define PD=Q;˛ .X/ to agree with the usual char-
acteristic polynomial when D is commutative and the reduced characteristic polynomial
when D has centre K  Q.
P ROPOSITION 10.23. Let K be a Q-subalgebra of End.A/ ˝ Q. (in particular, K and
End.A/˝Q have the same identity element), and assume that K is a field. Let f D ŒK W Q.
Then V` .A/ is a free K ˝Q Q` -module of rank .2 dim A/=f . Therefore, the trace of ˛ (as
an endomorphism of A) is .2g=f / TrK=Q .˛/ and deg.˛/ D NmK=Q .˛/2g=f .

P ROOF. In fact, we shall prove a stronger result in which D is assumed only to be a division
algebra (i.e., we allow it to be noncommutative). Let K be the centre of D, and let d D
                                                                                            2
   ŒD W K and f D ŒK W Q. If D˝Q Q` is again a division algebra, then V` .A/  V 2g=f d
p

where V is any simple D ˝Q Q` -module.13 In general, D ˝Q Q` will decompose into a
product                                            Y
                                     D ˝Q Q` D        Di
     13 Most of the theory of vector spaces over fields extends to modules over division algebras; in particular,

finitely generated modules have bases and so are free.
11. POLARIZATIONS AND INVERTIBLE SHEAVES                                                           53

                                                              Q
with each Di a simple algebra over Q` (if K ˝Q Q` D              Ki is the decomposition of
K ˝Q Q` into product ofLfields, then Di D D ˝K Ki ). Let Vi be a simple Mri .Di /-
module. Then V` .A/         i mi Vi for some mi  0. We shall that the mi are all equal.
Let ˛ 2 D. The characteristic polyomial P˛ .X/ of ˛ as an endomorphism of A is monic
of degree 2 dim A with coefficients in Q, and it is equal to the characteristic polynomial of
V` .˛/ acting on the Q` -vector space V` .A/. From the above decomposition of D ˝Q Q` ,
we find that                                 Y
                              PD=Q;˛ .X/ D      PDi =Q` ;˛ .X/:
From the isomorphism of D ˝Q Q` -modules V` .A/  ˚mi Vi , we find that
                                   Y
                          P˛ .X/ D    PDi =Q` ;˛ .X/mi :

If we assume that ˛ generates a maximal subfield of D, so that PD=Q;˛ .X/ is irreducible,
then the two equations show that any monic irreducible factor of P˛ .X/ in QŒX shares
a root with PD=Q;˛ .X /, and therefore equals it. Hence P˛ .X/ D PD=Q;˛ .X/m for some
integer m, and each mi equals m.                                                       2


C OROLLARY 10.24. Let ˛ 2 End0 .A/, and assume QŒ˛ is a product of fields. Let C˛ .X/
be the characteristic polynomial of ˛ acting on QŒ˛ (e.g., if QŒ˛ is a field, this is the
minimum polynomial of ˛); then

                           fa 2 C j C˛ .a/ D 0g D fa 2 C j P˛ .a/ D 0g:

N OTES . For an abelian variety A and an ` distinct from the characteristic of k, let A.`n / be
the `-primary component of the group A.k sep /, so A.`n / D lim A.k sep /.`n /. Thus A.`n / 
                                                             !n
.Q` =Z` /dim A . Then
                           Hom.A.`n /; B.`n // ' Hom.T` A; T` B/:
Most of the results in the section are due to Weil, but with A.`n / for T` A. Tate pointed out that it
was easier to work with the inverse limit T` A D lim A.k sep /.`n / rather than the direct limit, and
                                                      n
so T` A is called the Tate module of A. When asked whether it wouldn’t be more appropriate to
call it the Weil module, Weil responded that names in mathematics are like street names; they don’t
(necessarily) mean that the person had anything to do with it.14


11     Polarizations and Invertible Sheaves
As Weil pointed out, for many purposes, the correct higher dimensional analogue of an
elliptic curve is not an abelian variety, but a polarized abelian variety.
     A polarization15  of an abelian variety A is an isogeny W A ! A_ such that, over k al ,
 becomes of the form L for some ample sheaf L on Ak al . Unfortunately, this is not quite
the same as requiring that  itself be of the form L for L an ample invertible sheaf on A
(Milne 1986, 13.2).
     The degree of a polarization is its degree as an isogeny. An abelian variety together
with a polarization is called a polarized abelian variety. When  has degree 1, .A; / is
said to belong to the principal family, and  is called a principal polarization.
     There is the following very interesting theorem (Mumford 1970, p150).
   14 This was told to me first hand by David Gieseker.
   15 This notion of polarization differs slightly from Weil’s original definition.
54                                    CHAPTER I. ABELIAN VARIETIES: GEOMETRY


T HEOREM 11.1. Let L be an invertible sheaf on A, and let
                                 X
                        .L/ D       . 1/i dimk H i .A; L/
(Zariski cohomology).
  (a) The degree of L is .L/2 .
  (b) (Riemann-Roch) If L D L.D/, then .L/ D .D g /=gŠ.
  (c) If dim K.L/ D 0, then H r .A; L/ is nonzero for exactly one integer.

If L is ample, we know that dim K.L/ D 0. If L is very ample, we know that       .A; L/ ¤ 0,
and so the theorem implies that H r .A; L/ D 0 for all r ¤ 0.

A Finiteness Theorem
Up to isomorphism, there are only finitely many elliptic curves over a finite field k, because
each such curve can be realized as a cubic curve in P2 and there are only finitely many
cubic equations in three variables with coefficients in k. Using Theorem 10.1 it is possible
to extend this result to abelian varieties.
T HEOREM 11.2. Let k be a finite field, and let g and d be positive integers. Up to isomor-
phism, there are only finitely many abelian varieties A over k of dimension g possessing a
polarization of degree d 2 .

    Using Theorem 11.1, one shows that A can be realized as a variety of degree 3g d.gŠ/
     g
in P3 d 1 . The Chow form of such a variety is homogeneous of degree 3g d.gŠ/ in each of
g C 1 sets of 3g d variables, and it determines the isomorphism class of the variety. There
are only finitely many such polynomials with coefficients in k.
R EMARK 11.3. Theorem 11.2 played a crucial role in Tate’s proof of his conjecture (9.17)
over finite fields (see later).


12    The Etale Cohomology of an Abelian Variety
Let V be a variety over a field k. When k D C, we can endow V with the complex topology,
and form the cohomology groups H i .V; Q/. Weil was the first to observe that various
phenomena, for example the numbers of points on varieties, behaved as if the whole theory
(cohomology groups, Poincaré duality theorems, Lefschetz traces formula...) continued to
exist, even in characteristic p, and with the same Betti numbers. It is not clear whether Weil
actually believed that such a theory should exist, or that it just appeared to exist.
    However, there can’t exist cohomology groups with coefficients in Q and the correct
Betti numbers functorially attached to a variety in characteristic p because, if A is a super-
singular elliptic curve in characteristic p, then End0 .A/ a division algebra of dimension 4
over Q, and if H 1 .A; Q/ had dimension 2 over Q, then it would have dimension 1=2 over
End0 .A/, which is nonsense. However, End0 .A/  M2 .Q` /, ` ¤ p, and so there is no
reason there should not be a vector space H 1 .A; Q` / of dimension 2 over Q` functorially
attached to A — in fact, we know there is, namely V` A (better, its dual).
    Grothendieck constructed such a theory, and called it étale cohomology (see Milne 1980
or my notes LEC).
    For abelian varieties, the étale cohomology groups are what you would expect given the
complex groups.
12. THE ETALE COHOMOLOGY OF AN ABELIAN VARIETY                                                 55


T HEOREM 12.1. Let A be an abelian variety of dimension g over a separably closed field
k, and let ` be a prime different from char.k/.
  (a) There is a canonical isomorphism
                                                   '
                                   H 1 .Aet ; Z` / ! HomZ` .T` A; Z` /:
  (b) The cup-product pairings define isomorphisms
                              ^r
                                    H 1 .Aet ; Z` / ! H r .Aet ; Z` / for all r:

In particular, H r .Aet ; Z` / is a free Z` -module of rank 2g
                                                                  
                                                                r .

    If 1et .A; 0/ now denotes the étale fundamental group, then
                            H 1 .A; Z` / ' Homconts .1et .A; 0/; Z` /:
For each n, `nA W A ! A is a finite étale covering of A with group of covering transformations
Ker.`nA / D A`n .k/. By definition 1et .A; 0/ classifies such coverings, and therefore there
is a canonical epimorphism 1et .A; 0/ ! A`n .k/ (see Milne 1980, I 5). On passing to the
inverse limit, we get an epimorphism 1et .A; 0/  T` A, and consequently an injection
                                HomZ` .T` A; Z` / ,! H 1 .A; Z` /:
     To proceed further, we needL      to work with other coefficient groups. Let R be Z` ,
F` , or Q` , and write H .A/ for r0 H r .Aet ; R/. The cup-product pairing makes this
                            

into a graded, associative, anticommutative algebra. There is a canonical map H  .A/ ˝
H  .A/ ! H  .A  A/, which the Künneth formula shows to be an isomorphism when R
is a field. In this case, the addition map mW A  A ! A defines a map
                       m W H  .A/ ! H  .A  A/ ' H  .A/ ˝ H  .A/:
Moreover, the map a 7! .a; 0/W A ! A  A identifies H  .A/ with the direct summand
H  .A/ ˝ H 0 .A/ of H  .A/ ˝ H  .A/. As m ı .a 7! .a; 0// D id, the projection of
H  .A/ ˝ H  .A/ onto H  .A/ ˝ H 0 .A/ sends m .x/ to x ˝ 1. As the same remark applies
to a 7! .0; a/, this shows that
                                         X
             m .x/ D x ˝ 1 C 1 ˝ x C       xi ˝ yi ; deg.xi /; deg.yi / > 0:          (3)

L EMMA 12.2. Let H  be a graded, associative, anticommutative algebra over a perfect
field K. Assume that there is a map m W H  ! H  ˝ H  satisfying the identity (3). If
H 0 D K and H r D 0 for all r greater than some integer d , then dim.H 1 /  d , and when
equality holds, H  is canonically isomorphic to the exterior algebra on H 1 .

P ROOF. A fundamental structure theorem for Hopf algebras (Borel 1953) shows that H 
is equal to the associative algebra generated by certain elements xi subject only to the
relations imposed by the anticommutativity
                                     P              of H  and the nilpotence
                                                                            P of each xi . The
product of the xi has degree           deg.xi /, from which it follows that     deg.xi /  d . In
particular, the number of xi of degree 1 is  d ; as this number is equal to the dimension of
H 1 , this shows that its dimension is  d . When equality holds, all the xi must have degree
1; moreover, their squares must all be zero because otherwise there would be a nonzero
element x1 x2    xi2    xd of degree d C1. Hence H  is identified with the exterior algebra
on H 1 .                                                                                         2
56                                               CHAPTER I. ABELIAN VARIETIES: GEOMETRY


P ROOF ( OF T HEOREM 12.1) When R is Q` or F` , the conditions of the lemma are ful-
filled with d D 2g (Milne 1980, VI 1.1). Therefore H 1 .A; Q` / has dimension  2g. But
H 1 .A; Q` / ' H 1 .A; Z` / ˝Z` Q` , and so the earlier calculation shows that H 1 .A; Q` /
                                                                 r
                                    shows that H r .A; Q` / '      H 1 .A; Q` /, and, in par-
                                                               V
has dimension 2g. The lemma now
                               2g 
ticular, that its dimension if r . This implies that H .A; Z` / has rank 2g
                                                        r
                                                                            r . The exact
sequence (Milne 1980, VI 1.11)
                                     `                                                          `
              ! H r .A; Z` / ! H r .A; Z` / ! H r .A; F` / ! H rC1 .A; Z` / !   

now shows that dim H 1 .A; F` /  2g,
                                    and so the lemma implies that this dimension equals
2g and that dim H r .A; F` / D 2gr . On looking at the exact
                                                         Vrsequence    again, we see that
   r
H .A; Z` / must be torsion-free for all r. Consequently,     H .A; Z` / ! H 1 .A; Z` /
                                                                1

is injective because it becomes so when tensored with Q` , and it is surjective because it
becomes so when tensored with F` .                                                      2


R EMARK 12.3. In the course of the above proof, we have shown that the maximal abelian
`-quotient of 1et .A; 0/ is isomorphic to T` A. In fact, it is known that 1et .A; 0/ ' TA
                df
where TA D lim An .k/ (limit over the integers n > 0 prime to n). In order to prove this,
one has show that all finite étale coverings of A are isogenies.16 This is accomplished by
the following theorem (Mumford 1970, 18):
       Let A be an abelian variety over an algebraically closed field, and let f W A ! B
       be a finite étale covering with B connected; then it is possible to define on B
       the structure of abelian variety relative to which f is an isogeny.

R EMARK 12.4. We have shown that the following three algebras are isomorphic:
  (a) H  .A; Z` / with its cup-product structure;
  (b) the exterior algebra  H 1 .A; Z` / with its wedge-product structure;
                            V

  (c) the dual of  T` A with its wedge-product structure.
                  V

If we denote the pairing
                                            T` A  H 1 .A; Z` / ! Z`
by hji, then the pairing               ^r
                                               T` A  H r .A; Z` / ! Z`
is determined by
                             .a1 ^ : : : ^ ar ; b1 [ : : : [ br / D det.hai jbj i/:

R EMARK 12.5. Theorem 12.1 is still true if k is only separably closed (see Milne 1980, II
3.17). If A is defined over a field k, then the isomorphism
                                    ^
                                          Hom.T` ; Z` / ! H  .Ak sep ; Z` /

is compatible with the natural actions of Gal.k sep =k/.
     16 Actually, it suffices to show that all coverings are abelian. Perhaps this is easier.
13. WEIL PAIRINGS                                                                           57


13    Weil Pairings
For an elliptic curve A over a field k, and an integer m not divisible by the characteristic of
k, one has a canonical pairing

                              A.k al /m  A.k al /m ! m .k al /

where m .k al / is the group of mth roots of 1 in k al ( Z=mZ). This pairing is nondegen-
erate, skew-symmetric, and commutes with the action of
Gal.k al =k/.
    For an abelian variety A, this becomes a pairing

                           em W A.k al /m  A_ .k al /m ! m .k al /:

Again, it is nondegenerate, and it commutes with the action of
Gal.k al =k/. When combined with a polarization

                                          W A ! A_

this becomes a pairing
                            
                           em W Am .k al /  Am .k al / ! m .k al /,
                                             
                                            em .a; b/ D em .a; b/:

    The em -pairing can be defined as follows. For simplicity, assume k to be algebraically
                                                  0
closed. Let a 2 Am .k/ and let a0 2 A_                     0
                                     m .k/  Pic .A/. If a is represented by the divisor D
             
on A, then mA D is linearly equivalent to mD (see 8.5; mA is the map x 7! mxW A ! A),
which, by assumption, is linearly equivalent to zero. Therefore there are rational functions
f and g on A such that mD D .f / and mA D D .g/. Since

           div.f ı mA / D mA .div.f // D mA .mD/ D m.mA D/ D div.g m /,

we see that g m = .f ı mA / is rational function on A without zeros or poles — it is therefore
a constant function c on A. In particular,

                    g.x C a/m D cf .mx C ma/ D cf .mx/ D g.x/m :

Therefore g= .g ı ta / is a function on A whose mth power is one. This means that it is an
mth root of 1 in k.A/ and can be identified with an element of k. Define

                                  em .a; a0 / D g= .g ı ta / :

(Cf. Mumford 1970, 20, p184.)

L EMMA 13.1. Let m and n be integers not divisible by the characteristic of k. Then for all
a 2 Amn .k/ and a0 2 A_
                      mn .k/;

                                emn .a; a0 /n D em .na; na0 /:

P ROOF. See Milne 1986, 16.1 (or prove it as an exercise).                                   2
58                                          CHAPTER I. ABELIAN VARIETIES: GEOMETRY


    Let Z` .1/ D lim `n for ` a prime not equal to the characteristic of k. The lemma
allows us to define a pairing e` W T` A  T` A_ ! Z` .1/ by the rule17

                                    e` ..an /; .an0 // D .e`n .an ; an0 //:

For a homomorphism W A ! A_ , we define pairings
                          
                         em W Am  Am ! m .1/;              .a; a0 / 7! em .a; a0 /:
                        e` W T` A  T` A ! Z` .1/;        .a; a0 / 7! e` .a; a0 /:

If  D L , L 2 Pic.A/, then we write e`L for e` .
P ROPOSITION 13.2. There are the following formulas: for a homomorphism ˛W A ! B;
  (a) e` .a; ˛ _ .b// D e` .˛.a/; b/, a 2 T` A, b 2 T` BI
        ˛ _ ııf
  (b) e`         .a; a0 / D e` .˛.a/; ˛.a0 //, a; a0 2 T` A,  2 Hom.B; B _ /I
  (c) e`˛L .a; a0 / D e`L .˛.a/; ˛.a0 //, a; a0 2 T` A, L 2 Pic.B/.
      Moreover,
  (d) L 7! e`L is a homomorphism Pic.A/ ! Hom.2 T` A; Z` .1//; in particular, e`L is
      skew-symmetric.

P ROOF. See Milne 1986, 16.2 (or prove it as an exercise).                                                  2


E XAMPLE 13.3. Let A be an abelian variety over C. The exact sequence of sheaves on
A.C/
                                                       e 2 i./
                                   0 ! Z ! OAan           ! O
                                                             Aan ! 0
gives rise to an exact sequence

 H 1 .A.C/; Z/ ! H 1 .A.C/; O/ ! H 1 .A.C/; O / ! H 1 .A.C/; Z/ ! H 2 .A.C/; O/:

As

                                            H 1 .A.C/; O / ' Pic.A/ and
                             H 1 .A.C/; O/=H 1 .A.C/; Z/ ' A_ .C/;

we can extract from this an exact sequence

                          0 ! NS.A/ ! H 2 .A.C/; Z/ ! H 2 .A.C/; O/:

Let  2 NS.A/, and let E  be its image in H 2 .A.C/; Z/. Then E  can be regarded as a
skew-symmetric form on H1 .A.C/; Z/. It is a nondegenerate Riemann form if and only if
 is ample. As was explained above,  induces a pairing e` , and it is shown in Mumford
1970, p237, that the diagram

                            E  W H1 .A; Z/  H1 .A; Z/ !  Z
                                      #           #         #
                             
                            e` W    T` A       T` A    ! Z` .1/
     17 Note that e is ambiguous: it denotes both the Z=`Z pairing and the Z pairing. To avoid confusion, we
                   `                                                        `
shall use e` only to mean the Z` -pairing. Also, it should be noted that we sometimes write Z` .1/ multiplica-
tively and sometimes additively.
13. WEIL PAIRINGS                                                                         59


commutes with a minus sign if the maps H 1 .A.C/; Z/ ! T` A are taken to be the obvious
                                                                   n
ones and Z ! Z` .1/ is taken to be m 7!  m ,  D .: : : ; e 2 i=` ; : : :/. In other words,
                                                                    0
                                     e` .a; a0 / D  E .a;a / :

    In Milne 1986, 16, it is shown how étale cohomology can be used to give short proofs
of some important results concerning polarizations. For proofs that don’t use étale coho-
mology (and don’t neglect the p part in characteristic p) see Mumford 1970, 20, 23. Here
we merely list the results. We continue to assume that k is algebraically closed.

T HEOREM 13.4. Let ˛W A ! B be an isogeny of degree prime to the characteristic of k,
and let  2 NS.A/. Then  D ˛  .0 / for some 0 2 NS.B/ if and only if, for all `
dividing deg.˛/, there exists a skew-symmetric form eW T` B  T` B ! Z` .1/ such that
e` .a; a0 / D e.˛.a/; ˛.a0 // for all a; a0 2 T` A.

C OROLLARY 13.5. Assume ` ¤ char.k/.                                              n
                               n
                                        V2 An element  of NS.A/ is divisible by ` if
and only if e` is divisible by ` in Hom.   T` A; Z` .1//:

P ROOF. Apply the proposition to `nA W A ! A.                                              2


P ROPOSITION 13.6. Assume char.k/ ¤ 2; `. A homomorphism W A ! A_ is of the
form L for some L 2 Pic.A/ if and only if e` is skew-symmetric.

L EMMA 13.7. Let P be the Poincaré sheaf on A  A_ . Then

                          e`P ..a; b/; .a0 ; b 0 // D e` .a; b 0 /       e` .a0 ; b/

for a; a0 2 T` A and b; b 0 2 T` A_ .

    For a polarization W A ! A_ , define

                                  e  W Ker./  Ker./ ! m

as follows: suppose m kills Ker./, and let a and a0 be in Ker./; choose a b such that
mb D a0 , and let e  .a; a0 / D em .a; b/; this makes sense because m.b/ D .mb/ D 0.
Also it is independent of the choice of b and m because if mnb 0 D a0 and nc D a, then

                                   emn .a; b 0 / D emn .c; b/n
                                                   D em .a; nb 0 /

(by 13.1), and so

                        emn .a; b 0 /=em .a; b/ D em .a; .nb 0                b//
                                                          
                                                       D em .a; nb 0 b/
                                                          
                                                       D em .nb 0 b; a/ 1
                                                       D1
60                                   CHAPTER I. ABELIAN VARIETIES: GEOMETRY


because a D 0. Let a D .an / and a0 D .an0 / be in T` A. If am D 0 D am
                                                                         0 for some m,

then

                            e  .am ; am
                                       0                 0
                                         / D e`m .am ; a2m /
                                                                   m
                                                           0
                                           D e`2m .a2m ; a3m /`
                                           D e`2m .a2m ; a2m
                                                           0
                                                              /:

Note that this implies that e  is skew-symmetric.

P ROPOSITION 13.8. Let ˛W A ! B be an isogeny of degree prime to char.k/, and let
W A ! A_ be a polarization of A. Then  D ˛  .0 / for some polarization 0 on B if and
only if Ker.˛/  Ker./ and e  is trivial on Ker.˛/  Ker.˛).

R EMARK 13.9. The degrees of  and 0 are related by

                               deg./ D deg.0 /  deg.˛/2 ;

because  D ˛ _ ı 0 ı ˛.

C OROLLARY 13.10. Let A be an abelian variety having a polarization of degree prime to
char.k/. Then A is isogenous to a principally polarized abelian variety.

P ROOF. Let  be a polarization of A, and let ` be a prime dividing the degree of . Choose
a subgroup N of Ker./ of order `, and let B D A=N . As e  is skew-symmetric, it must
be zero on N  N , and so the last proposition implies that B has a polarization of degree
deg./=`2 .                                                                               2


C OROLLARY 13.11. Let  be a polarization of A, and assume that Ker./  Am with m
prime to char.k/. If there exists an element ˛ of End.A/ such that ˛.Ker.//  Ker./
and ˛ _ ı  ı ˛ D  on Am2 , then A  A_ is principally polarized.

   Since .A  A_ /_ ' A_  A, there is a canonical isomorphism A  A_ ! .A  A_ /_ .
The problem is to show that there is such an isomorphism that is a polarization, i.e., of the
form L for some ample invertible sheaf L — for this we need to replace A with A4 .

T HEOREM 13.12 (Z ARHIN ’ S TRICK ). For any abelian variety A, .A  A_ /4 is principally
polarized.

P ROOF. To prove the theorem, we have to prove that, for every polarized abelian variety
.A; /, there exists an ˛ satisfying the condition of (13.11) for .A4 ; 4 /. Lagrange showed
that every positive integer is a sum of 4 squares (ANT 4.20). Therefore, there are integers
a; b; c; d such that
                            a2 C b 2 C c 2 C d 2  1 mod m2 ;
and we let              0                        1
                       a         b     c       d
                     B b        a     d        c C
                   ˛DB                           C 2 M4 .Z/  End.A4 /:
                     @ c         d    a       b A
                       d        c      b      a
14. THE ROSATI INVOLUTION                                                                         61


Since ˛ commutes with 4 D diag.; ; ; /, we have

                                    ˛.Ker.4 //  Ker.4 /:

Moreover, ˛ _ is the transpose of ˛ (as a matrix), and so

                          ˛ _ ı 4 ı ˛ D ˛ tr ı 4 ı ˛ D 4 ı ˛ tr ı ˛:

But
                              ˛ tr ı ˛ D .a2 C b 2 C c 2 C d 2 /I4 :                               2



C OROLLARY 13.13. Let k be a finite field; for each integer g, there exist only finitely
many isomorphism classes of abelian varieties of dimension g over k.

P ROOF. Let A be an abelian variety of dimension g over k. From (13.12) we know that
.A  A_ /4 has a principal polarization, and according to (11.2), the abelian varieties of
dimension 8g over k having principal polarizations fall into finitely many isomorphism
classes. But A is a direct factor of .A  A_ /4 , and (15.1) shows that .A  A_ /4 has only
finitely many direct factors.                                                             2


N OTES . Corollary 13.5 is “the proposition on the last page of Weil 1948b” that plays a crucial role
in Tate 1966 (ibid. p137). In the proof of Tate’s theorem, it can now be replace by “Zarhin’s trick”
which, however, depends on the stronger form of the Corollary 13.5.


14     The Rosati Involution
                                                                                                  df
Fix a polarization  on A. As  is an isogeny A ! A_ , it has an inverse in Hom0 .A_ ; A/ D
Hom.A_ ; A/ ˝ Q. The Rosati involution on End.A/ ˝ Q corresponding to  is

                                    ˛ 7! ˛  D  1 ı ˛ _ ı :

This has the following obvious properties:

                 .˛ C ˇ/ D ˛  C ˇ  ;      .˛ˇ/ D ˇ˛; a D a for a 2 Q:

For any a; a0 2 T` A ˝ Q, ` ¤ char.k/,

                e` .˛a; a0 / D e` .˛a; a0 / D e` .a; ˛ _ ı a0 / D e` .a; ˛  a0 /;

from which it follows that ˛  D ˛.

R EMARK 14.1. The second condition on ˛ in (13.11) can now be stated as ˛  ı ˛ D                  1
on Am2 (provided ˛  lies in End.A/).

P ROPOSITION 14.2. Assume that k is algebraically closed. Then the map

                        L 7!  1 ı L W NS.A/ ˝ Q ! End.A/ ˝ Q;

identifies NS.A/ ˝ Q with the subset of End.A/ ˝ Q of elements fixed by .
62                                      CHAPTER I. ABELIAN VARIETIES: GEOMETRY


P ROOF. Let ˛ 2 End0 .A/, and let ` be an odd prime distinct from the characteristic of k.
According to (13.6),  ı ˛ is of the form L if and only if e`ı˛ .a; a0 / D e`ı˛ .a0 ; a/ for
all a; a0 2 T` A ˝ Q. But

                              e`ı˛ .a; a0 / D e` .a; ˛a0 /
                                            D    e` .˛a0 ; a/
                                            D    e` .a0 ; ˛ _ ı .a//;

and so this is equivalent to  ı ˛ D ˛ _ ı , i.e., to ˛ D ˛  .                             2


      In general, this set will not be a subalgebra of End.A/ ˝ Q, because ˛ and ˇ can
       be fixed by  without ˛ˇ being fixed (when ˛ and ˇ are fixed, then .˛ˇ/ D ˇ˛,
which need not equal ˛ˇ).

     The next result is very important.
T HEOREM 14.3. The bilinear form

                  .˛; ˇ/ 7! Tr.˛ ı ˇ  /W End.A/ ˝ Q  End.A/ ˝ Q!Q

is positive definite, i.e., Tr.˛ ı ˛  / > 0 for ˛ ¤ 0. More precisely, let D be the ample
divisor defining the polarization used in the definition of ; then
                                              2g
                            Tr.˛ ı ˛  / D     g
                                                  .D g 1  ˛  .D//:
                                             .D /

P ROOF. As D is ample and ˛  .D/ is effective, the intersection number .D g 1  ˛  .D// is
positive. Thus the second statement implies the first. The second statement is proved by a
calculation, which we omit for the present (see Milne 1986, 17).                         2


P ROPOSITION 14.4. Let  be a polarization of the abelian variety A.
  (a) The group of automorphisms of .A; / is finite.
  (b) For any integer n  3, an automorphism of .A; / acting as the identity on An .k al /
      is equal to the identity.

P ROOF. (a) Let ˛ be an automorphism of A. In order for ˛ to be an automorphism of
.A; /, we must have  D ˛ _ ı  ı ˛, and therefore ˛  ˛ D 1, where  is the Rosati
involution defined by . Consequently,

                    ˛ 2 End.A/ \ f˛ 2 End.A/ ˝ R j Tr.˛  ˛/ D 2gg:

But End.A/ is discrete in End.A/ ˝ R, and End.A/ ˝ R is compact.
     (b) Assume further that ˛ acts as the identity on An . Then ˛ 1 is zero on An , and so
it is of the form nˇ with ˇ 2 End.A/ (see 10.16). The eigenvalues of ˛ and ˇ are algebraic
integers, and those of ˛ are roots of 1 because it has finite order. The next lemma shows
that the eigenvalues of ˛ equal 1. Therefore ˛ is unipotent, and so ˛ 1 D nˇ is nilpotent.
                                df
Suppose that ˇ ¤ 0. Then ˇ 0 D ˇ  ˇ ¤ 0 because Tr.ˇ  ˇ/ > 0. As ˇ 0 D ˇ 0 , this implies
that Tr.ˇ 02 / > 0, and so ˇ 02 ¤ 0. Similarly, ˇ 04 ¤ 0, and so on, which contradicts the
nilpotence of ˇ.                                                                           2
15. GEOMETRIC FINITENESS THEOREMS                                                           63


L EMMA 14.5. If  is a root of 1 such that for some algebraic integer      and integer n  3,
 D 1 C n , then  D 1.

P ROOF. If  ¤ 1, then, after raising it to a power, we may assume that it is a primitive pth
root of 1 for some prime p. Then NmQŒ =Q . 1/ D . 1/p 1 p, because the minimum
polynomial of  1 is

                         .X C 1/p 1 C    C 1 D X p 1 C    C p

(see ANT, Chapter 6). Now the equation       1 D n implies that . 1/p 1 p D np 1 N. /.
This is impossible because p is prime.                                                       2


R EMARK 14.6. (a) Let .A; / and .A0 ; 0 / be polarized abelian varieties over a perfect
field k. If there exists an n  3 such that both A and A0 have all their points of order n
rational over k, then any isomorphism .A; / ! .A; 0 / defined over some extension field
of k is automatically defined over k (apply AG 16.9).
     (b) Let ˝  k be fields such that the fixed field of       D Aut.˝=k/ is k and ˝ is
algebraically closed. Let .A; / be a polarized abelian variety over ˝, and let .' / 2
be a descent system on .A; /, i.e., ' is an isomorphism .A; / ! .A; / and ' ı
. ' / D ' for all ;  2 . Assume that for some n  3, there exists a subfield K of ˝
finitely generated over k such that ' .P / D P for all P killed by n and all  fixing K.
Then .A; / has a model over k splitting .' /, i.e., there exists a polarized abelian variety
.A0 ; 0 / over k and an isomorphism 'W V0˝ ! V such that ' D ' 1 ı ' for all  2
(apply AG 16.33).


15     Geometric Finiteness Theorems
In this section we prove two finiteness theorems that hold for abelian varieties over any field
k. The first theorem says that an abelian variety can be endowed with a polarization of a
fixed degree in only a finite number of essentially different ways. The second says that,
up to isomorphism, an abelian variety has only finitely many direct factors. As a corollary
we find that there are only finitely many isomorphism classes of abelian varieties of a fixed
dimension over a finite field. This simplifies the proof of Tate’s isogeny theorem.

T HEOREM 15.1. Let A be an abelian variety over a field k, and let d be an integer; then
there exist only finitely many isomorphism classes of polarized abelian varieties .A; / with
 of degree d .

     Let .A; / and .A0 ; 0 / be polarized abelian varieties. From a homomorphism ˛W A !
A0 , we obtain a map
                                      df
                              ˛  .0 / D ˛ _ ı 0 ı ˛W A ! A_ :
When ˛ is an isomorphism and ˛  .0 / D , we call ˛ an isomorphism .A; / ! .A0 ; 0 /
of polarized abelian varieties.
    The theorem can be restated as follows: Let Pol.A/ be the set of polarizations on A,
and let End.A/ act on Pol.A/ by u 7! u_ ı  ı u; then there are only finitely many orbits
under this action.
64                                    CHAPTER I. ABELIAN VARIETIES: GEOMETRY


    Note that End.A/ D Aut.A/. If u is an automorphism of A, and L is an ample
invertible sheaf on A, the u L is also an ample invertible sheaf, and u L D u_ ı L ı u;
thus End.A/ does act on Pol.A/.
    Fix a polarization 0 of A, and let  be the Rosati involution on End.A/ ˝ Q defined
by 0 . The map  7! 0 1 ı  identifies Pol.A/ with a subset of the set .End.A/ ˝ Q/
of elements of End.A/ ˝ Q fixed by . Because 0 is an isogeny, there exists an isogeny
˛W A_ ! A such that ˛ ı 0 D n, some n 2 Z, and then 0 1 D .nA1 / ı ˛. Therefore the
image of
                                Pol.A/ ,! .End.A/ ˝ Q/
          df
lies in L D n 1 End.A/.
     Let End.A/ act on End.A/ ˝ Q by
                   ˛ 7! u ı ˛ ı u, u 2 End.A/ , ˛ 2 End.A/ ˝ Q:
Then L is stable under this action, and the map Pol.A/ ! End.A/ ˝ Q is equivariant for
this action, because u_ ı  ı u 7! 0 1 ı u_ ı  ı u D 0 1 ı u_ ı 0 ı 0 1 ı  ı u D
u ı .0 1 / ı u:
    Note that deg.0 1 ı / D deg.0 / 1 deg./. Also (see 10.23), for an endomorphism
˛ of A, deg.˛/ is a fixed power of Nm.˛/ (norm from End.A/ ˝ Q to Q/. Therefore, as
 runs through a subset of Pol.A/ of elements with bounded degrees, then 0 1 ı  runs
through a subset of L of elements with bounded norms. Thus the theorem is a consequence
of the following number theoretic result.
P ROPOSITION 15.2. Let E be a finite-dimensional semisimple algebra over Q with an
involution , and let R be an order in E. Let L be a lattice in E that is stable under the
action ˛ 7! u ˛u of R on E. Then for any integer N , there are only finitely many orbits
for the action of R on
                              S D fv 2 L j Nm.v/  N g;
i.e., S=R is finite.

    An order in E is a subring R of E that is a full lattice, i.e., free of rank dim.E/ over Z.
In the application, R D End.A/.
    This proposition will be proved using a general result from the reduction theory of
arithmetic subgroups — see below (15.9).
    We come now to the second main result of this section. An abelian subvariety B of A
is said to be a direct factor of A if there exists an abelian subvariety C of A such that the
map .b; c/ 7! b C cW B  C ! A is an isomorphism. Two direct factors B and B 0 of A are
said to be isomorphic if there exists an automorphism ˛ of A such that ˛.B/ D B 0 .
T HEOREM 15.3. Up to isomorphism, an abelian variety A has only finitely many direct
factors.

P ROOF. Let B be a direct factor of A with complement C say, and define e to be the
composite
                                         .b;c/7!.b;0/
                           A'B C                  ! B  C ' A:
Then e is an idempotent (i.e., e 2 D e), and B is determined by e because B ' Ker.1         e/.
Conversely, for any idempotent e of End.A/
                                 A D Ker.1      e/  Ker.e/:
15. GEOMETRIC FINITENESS THEOREMS                                                          65


Let u 2 End.A/ . Then e 0 D ueu 1 is also an idempotent in End.A/, and u defines an
isomorphism
                            Ker.1 e/ ! Ker.1 e 0 /:
Therefore, we have a surjection

            fidempotents in End.A/g= End.A/ ! fdirect factors of Ag= ;

and so the theorem is a consequence of the following number theoretic result.               2


P ROPOSITION 15.4. Let E be a semisimple algebra of finite dimension over Q, and let R
be an order in E. Then
                             fidempotents in Rg=R
is finite (here R acts on the set of idempotents by conjugation).

    This proposition will again be proved using a general result from the reduction theory
of arithmetic subgroups (see 15.8), which we now state.

T HEOREM 15.5. Let G be a reductive group over Q, and let be an arithmetic subgroup
of G.Q/; let G ! GL.V / be a representation of G on a Q-vector space V , and let L be a
lattice in V that is stable under . If X is a closed orbit of G in V , then L \ X is the union
of a finite number of orbits of .

P ROOF. Borel 1969, 9.11. (The theorem is due to Borel and Harish-Chandra, but special
cases of it were known earlier.)                                                    2


R EMARK 15.6. (a) By an algebraic group we mean an affine group variety. It is reductive
if it has no closed normal connected subgroup U consisting of unipotent elements (i.e.,
elements such that un D 1 for some n). A connected algebraic group G is reductive if and
only if the identity component Z 0 of its centre is a torus and G=Z 0 is a semisimple group.
For example, GLn is reductive. The group
                                       ˚     ˇ
                                 B D a0 bc ˇ ac ¤ 0
                                ˚     
is not reductive, because U D 10 b1 is a closed normal connected subgroup consisting
of unipotent matrices.
     (b) Let G be an algebraic group over Q. Then G can be realized as a closed subgroup
of GLn .Q/ for some n (this is often taken to be the definition of an algebraic group). Let

                        GLn .Z/ D fA 2 Mn .Z/ j det.A/ D ˙1g:

Then GLn .Z/ is a group, and we let 0 D GLn .Z/ \ G.Q/. A subgroup of G.Q/ is
said to be arithmetic if it is commensurable with 0 , i.e., if \ 0 is of finite index in both
   and 0 — this is an equivalence relation. One can show that, although 0 depends on
the choice of the embedding G ,! GLn , two embeddings give commensurable groups, and
hence the notion of an arithmetic subgroup doesn’t depend on the embedding. Let

                   .N / D fA 2 G.Q/ j A 2 Mn .Z/;         A  I mod.N /g:
66                                         CHAPTER I. ABELIAN VARIETIES: GEOMETRY


Then .N / is a subgroup of finite index in 0 , and so it is arithmetic. An arithmetic
subgroup of this type is said to be a principal congruence subgroup. 18
    (c) A representation of G on a vector space V is a homomorphism G ! GL.V / of
algebraic groups. We can regard V itself as an algebraic variety (the choice of a basis for V
determines an isomorphism V  An , n D dim.V //, and we are given mapping of algebraic
varieties
                                         G V !V.
      If v is an element of V , then the orbit Gv is the image of the map

                                      G  fvg ! V , g 7! g  v:

It is a constructible set, but it need not be closed in general. To check that the orbit is closed,
one needs to check that
                                    X.k al / D fgv j g 2 G.k al /g
is closed in V ˝ k al . An /. One should interprete L \ X as L \ X.k al /.

      We give three applications of (15.5).

A PPLICATION 15.7. Let G D SLn , and let D SLn .Z/. Then G acts in a natural way on
the space V of quadratic forms in n variables with rational coefficients,
             X
       V Df      aij Xi Xj j aij 2 Qg D fsymmetric n  n matrices, coeffs in Qg;

— if q.X / D XAX tr , then .gq/.X / D X  gAg tr  X tr — and preserves the lattice L of
such forms with integer coefficients. Let q be a quadratic form with nonzero discriminant
d , and let X be the orbit of q, i.e., the image G  q of G under the map of algebraic
varieties g 7! g  qW G ! V . The theory of quadratic forms shows that X.Qal / is equal
to the set of all quadratic forms (with coefficients in Qal / of discriminant d . Clearly this is
closed, and so the theorem shows that X \ L contains only finitely many SLn .Z/-orbits:
the quadratic forms with integer coefficients and discriminant d fall into a finite number of
proper equivalence classes.

A PPLICATION 15.8. With the notations of (15.4), there exists an algebraic group G over
Q with G.Q/ D E  which is automatically reductive (this only has to be checked over Qal ;
but E ˝ Qal is a product of matrix algebras, and so GQal is a product of GLn s). Take to
be the arithmetic subgroup R of G.Q/, V to be E with G acting by inner automorphisms,
and L to be R. Then the idempotents in E form a finite set of orbits under G, and each
of these orbits is closed. In proving these statements we may again replace Q by Qal and
assume E to be a product of matrix algebra; in fact, we may take E D Mn .k/. Then the
argument in the proof of (15.3) shows that

                     fidempotents in Eg=E  ' fdirect factors of k n g=  :

But, up to isomorphism, there is only one direct factor of k n for each dimension  n.
Thus, each idempotent is conjugate to one of the form e D diag.1; :::; 1; 0; :::; 0/. If r
is the number of 1s, then the orbit of e under E  corresponds to the set of subspaces of
     18 The congruence subgroup problem asks whether every arithmetic subgroup contains a congruence sub-

group. It has largely been solved — for some groups G they do; for some groups G they don’t.
16. FAMILIES OF ABELIAN VARIETIES                                                           67


k n of dimension r. The latter is a Grassmann variety, which is complete (e.g., the orbit
of e D diag.1; 0; :::; 0/ corresponds to the set of lines in k n through the origin, i.e., with
Pn /, and hence is closed when realized as a subvariety of any variety. Now we can apply
Theorem 15.5 to obtain Proposition 15.4.

A PPLICATION 15.9. With the notations of (15.2), let G be the algebraic group over Q such
that
                         G.Q/ D fa 2 E j Nm.a/ D ˙1g;
let   D R , let V D E, and let L  V the lattice in (5.2). One verifies:
  (a) G is a reductive group having as an arithmetic subgroup;
  (b) the orbits of G on V are all closed;
                                        df
  (c) for any rational number d , Vd D fv 2 V j Nm.v/ D d g is the union of a finite
      number of orbits of G.
Then (15.5) shows that L \ Vd comprises only finitely many          -orbits, as is asserted by
(15.2). For details, see Milne 1986, 18.


16    Families of Abelian Varieties
Let S be a variety over k. A family of abelian varieties over S is a proper smooth map
W A ! S with a law of composition

                                     multW A S A ! A

and a section such that each fibre is an abelian variety. A homomorphism of families of
abelian varieties is a regular map ˛W A ! B compatible with the structure maps A ! S ,
B ! S, and with the multiplication maps. Many results concerning abelian varieties extend
to families of abelian varieties.

P ROPOSITION 16.1 (R IGIDITY L EMMA ). Let S be a connected variety, and let W V ! S
be a proper flat map whose fibres are geometrically connected varieties; let  0 W V 0 ! S be
a variety over S , and let ˛W V ! V 0 be a morphism of varieties over S . If for some point
s of S, the image of the fibre Vs in Vs0 is a single point, then f factors through S (that is,
there exists a map f 0 W S ! V 0 such that f D f 0 ı ).

P ROOF. Mumford, D., Geometric Invariant Theory, Springer, 1965, 6.1.                        2


C OROLLARY 16.2. (a) Every morphism of families of abelian varieties carrying the zero
section into the zero section is a homomorphism.
    (b) The group structure on a family of abelian varieties is uniquely determined by the
choice of a zero section.
    (c) A family of abelian varieties is commutative.

P ROOF. (a) Apply the proposition to the map 'W A  A ! B defined as in the proof of
(1.2).
    (b) This follows immediately from (a).
    (c) The map a 7! a 1 is a homomorphism.                                       2
68                                     CHAPTER I. ABELIAN VARIETIES: GEOMETRY


    The next result is a little surprising: it says that a constant family of abelian varieties
can’t contain a nonconstant subfamily.
P ROPOSITION 16.3. Let A be an abelian variety over a field k, and let S be a variety over
k such that S.k/ ¤ ;. For any injective homomorphism ˛W B ,! A  S of families of
abelian varieties over S , there is an abelian subvariety B of A (defined over k/ such that
˛.B/ D B  S .

P ROOF. Let s 2 S.k/, and let B D Bs (fibre over s/. Then ˛s identifies B with a subvariety
of A. The map hW B ,! A  S  .A=B/  S has fibre Bs ! A ! A=Bs over s, which
is zero, and so (16.1) shows that h D 0. It follows that ˛.B/  B  S. In fact, the two are
equal because their fibres over S are connected and have the same dimension.              2


   Recall that a finitely generated extension K of a field k is regular if it is linearly disjoint
from k al ; equivalently, if K D k.V / for some variety V over k.
C OROLLARY 16.4. Let K be a regular extension of a field k.
  (a) Let A be an abelian variety over k. Then every abelian subvariety of AK is defined
      over k.
  (b) If A and B are abelian varieties over k, then every homomorphism ˛W AK ! BK is
      defined over k.

P ROOF. (a) Let V be a variety over k such that k.V / D K. After V has been replaced
by an open subvariety, we can assume that B extends to a family of abelian varieties over
V . If V has a k-rational point, then the proposition shows that B is defined over k. In
any case, there exists a finite Galois extension k 0 of k and an abelian subvariety B 0 of Ak 0
             0                                                                             0
such that BKk   0 D BKk 0 as subvarieties of AKk 0 . The equality uniquely determines B as
a subvariety of A. As B has the same property for any  2 Gal.k 0 =k/, we must have
B D B, and this shows that B is defined over k.
    (b) Part (a) shows that the graph of ˛ is defined over k.                                2


T HEOREM 16.5. Let K=k be a regular extension of fields, and let A be an abelian variety
over K. Then there exists an abelian variety B over k and a homomorphism ˛W BK !
A with finite kernel having the following universal property: for any abelian variety B 0
                          0
and homomorphism ˛ 0 W BK   ! A with finite kernel, there exists a unique homomorphism
    0                 0
'W B ! B such that ˛ D ˛ ı 'K .

P ROOF. Consider the collection of pairs .B; ˛/ with B an abelian variety over k and ˛
a homomorphism BK ! A with finite kernel, and let A be the abelian subvariety of
A generated by the images the ˛. Consider two pairs .B1 ; ˛1 / and .B2 ; ˛2 /. Then the
identity component C of the kernel of .˛1 ; ˛2 /W .B1  B2 /K ! A is an abelian subvariety
of B1  B2 , which (16.4) shows to be defined over k. The map .B1  B2 =C /K ! A
has finite kernel and image the subvariety of A generated by ˛1 .B1 / and ˛2 .B/. It is now
clear that there is a pair .B; ˛/ such that the image of ˛ is A . Divide B by the largest
subgroup scheme N of Ker.˛/ to be defined over k. Then it is not difficult to see that
                                                                   0
the pair .B=N; ˛/ has the correct universal property (given ˛ 0 W BK ! A, note that for a
                                                     0
suitable C contained in the kernel of .B=N /K  BK ! A, the map b 7! .b; 0/W B=N !
.B=N /  B 0 =C is an isomorphism).                                                       2
17. NÉRON MODELS; SEMISTABLE REDUCTION                                                      69


R EMARK 16.6. The pair .B; ˛/ is obviously uniquely determined up to a unique isomor-
phism by the condition of the theorem; it is called the K=k-trace of A. (For more details
on the K=k-trace and the reverse concept, the K=k-image, see Lang 1959, VIII; Conrad
2006; Kahn 2006, App. A).

Mordell-Weil theorem.
Recall that, for an elliptic curve A over a number field K, A.K/ is finitely generated (Milne
2006, Chapter IV). More generally, there is the following theorem.
T HEOREM 16.7. Let A be an abelian variety over a field K that is finitely generated over
the prime field. Then A.K/ is finitely generated.

P ROOF. For elliptic curves over Q, this was proved by Mordell; for Jacobian varieties, it
was proved by Weil in his thesis (Weil stated his result in terms of curves, since Jacobian
varieties hadn’t been defined over any fields except C at the time); it was proved for abelian
varieties over number fields by Taniyama; the extension to other fields was made by Lang
and Néron. For a proof for abelian varieties over number fields, see Serre 1989, Chapter
4.                                                                                           2


   Clearly, one needs some condition on K in order to have A.K/ finitely generated — if
K is algebraically closed, A.K/tors is never finitely generated. However, Lang and Néron
(1959) proved the following result.
T HEOREM 16.8. Assume K is finitely generated (as a field) over k, and that k is alge-
braically closed in K. Let .B; ˛/ be the K=k trace of A. Then A.K/=˛.B/.k/ is finitely
generated.

     For proofs, see Conrad 2006; Hindry (App. B to Kahn 2006); Kahn ArXive:math.AG/0703063.


17     Néron models; Semistable Reduction
Let R be a discrete valuation ring with field of fractions K and residue field k. Let  be
a prime element of R, so that k D R=./. We wish to study the reduction19 of an elliptic
curve E over K. For simplicity, we assume p ¤ 2; 3. Then E can be described by an
equation
                                                  df
                     Y 2 D X 3 C aX C b;  D 4a3 C 27b 2 ¤ 0:
By making the substitutions X 7! X=c 2 , Y 7! Y =c 3 , we can transform the equation to
                                      Y 2 D X 3 C ac 4 X C bc 6 ;
and this is essentially the only way we can transform the equation. A minimal equation for
E is an equation of this form with a; b 2 R for which ord./ is a minimum. A minimal
equation is unique up to a transformation of the form
                                    .a; b/ 7! .ac 4 , bc 6 /, c 2 R :
   Choose a minimal equation for E, and let E0 be the curve over k defined by the equation
mod ./. There are three cases:
  19 For another discussion of the Néron models of elliptic curves, see Milne 2006, II 6.
70                                            CHAPTER I. ABELIAN VARIETIES: GEOMETRY


  (a) E0 is nonsingular, and is therefore is an elliptic curve. This occurs when ord./ D 0.
      In this case, we say that E has good reduction.
  (b) E0 has a node. This occurs when j but does not divide both a and b. In this
                          df
      case E0 .k/nonsing D E0 .k/ fnodeg is isomorphic to k  as an algebraic group (or
      becomes so after a quadratic extension of k), and E is said to have multiplicative
      reduction.
  (c) E0 has a cusp. This occurs when  divides both a and b (and hence also /. In this
      case E0 .k/nonsing is isomorphic to k C , and E is said to have additive reduction.
    The curve E is said to have semistable reduction when either (a) or (b) occurs. Now
suppose we extend the field from K to L, ŒL W K < 1, and choose a discrete valuation ring
S with field of fractions L such that S \ K D R. When we pass from K to L, the minimal
equation of E remains minimal in cases (a) and (b), but it may change in case (c). For a
suitable choice of L, case (c) will become either case (a) or case (b). In other words, if E
has good reduction (or multiplicative) reduction over K, then the reduction stays good (or
multiplicative) over every finite extension L; if E has additive reduction, then the reduction
can stay additive or it may become good or multiplicative over an extension L, and for a
suitable extension it will become good or multiplicative.
    The proof of the statement is elementary. For example, suppose E has additive reduc-
tion, and adjoin a sixth root $ of  to K. Then we can replace the equation by

                                   Y 2 D X 3 C .a=$ 4 /X C .b=$ 6 /:

If both ordL .a=$ 4 / > 0 and ordL .b=$ 6 / > 0, then continue....
    These statements extend to abelian varieties, but then become much more difficult to
prove.
T HEOREM 17.1 (N ÉRON ). Let A be an abelian variety over a field K as above. Then there
is a canonical way to attach to A an algebraic group A0 over k.

R EMARK 17.2. In fact Néron proves the following: the functor from smooth schemes over
R,
                       S 7! HomSpec K .S Spec R Spec K; A/
is representable by a smooth group scheme A over R. The scheme A is unique (because
of the Yoneda lemma–see AG 1.39), and we set A0 D A Spec R Spec k. The scheme A is
called the Néron model of A.

      A general theorem on algebraic groups shows that A0 has a filtration:

                                         A0  .A0 /0  .A0 /1  0

with A0 =.A0 /0 a finite algebraic group (.A0 /0 is the connected component of A0 containing
the identity element), .A0 /0 =.A0 /1 an abelian variety, and .A0 /1 a commutative affine
group scheme. Again there are three cases to consider:
  (a) A0 is an abelian variety. In this case A is said to have good reduction.
  (b) .A0 /1 is a torus20 , i.e., after a finite extension of k, .A0 /1 becomes isomorphic to a
      product of copies A1 f0g D k  .
     20 This notion should not be confused with that of a complex torus discussed in 2.
18. ABEL AND JACOBI                                                                                         71


  (c) .A0 /1 contains copies of A1 D k C .
     The abelian variety A is said to have semistable reduction in case (a) or (b).

T HEOREM 17.3. If A has good reduction, then A0 doesn’t change under a finite field exten-
sion; if A has semistable reduction, then .A0 /0 doesn’t change under a finite field extension;
A always acquires semistable reduction after a finite extension.

    The proofs of these theorems are long and quite difficult. Fortunately, for most purposes
one only needs the statements, and these are very believable given what is true for elliptic
curves
N OTES . The original paper of Néron (1964) is almost unreadable, because it is written in a private
language (a relative version of Weil’s language).21 The article Artin 1986 is too concise. In view of
this, the book Bosch et al. 1990 is invaluable. It gives a very complete and detailed treatment of the
topic.


18     Abel and Jacobi
A BEL 1802-29.
JACOBI 1804-1851.
   Let f .X; Y / 2 RŒX; Y . We can regard the equation f .X; Y / D 0 as defining Y
(implicitly) as a multivalued function of X . An integral of the form,
                                        Z
                                           g.Y /dX

with g.Y / a rational function, is called an abelian integral, after Abel who made a profound
study of them. For example, if f .X; Y / D Y 2 X 3 aX b, then

                                   dX                 dX
                               Z           Z
                                        D                       1
                                    Y            3
                                              .X C aX C b/ 2
is an example of an abelian integral — in this case it is a elliptic integral, which had been
studied in the eighteenth century. The difficulty with these integrals is that, unless the curve
f .X; Y / D 0 has genus 0, they can’t be evaluated in terms of the elementary functions.
    Today, rather than integrals of multivalued functions, we prefer to think of differentials
on a Riemann surface, e.g., the compact Riemann surface (i.e., curve over C) defined by
f .X; Y / D 0.
    Let C be a compact Riemann surface. Recall Cartan 1963 that C is covered by coor-
dinate neigbourhoods .U; z/ where U can be identified with an open subset of C and z is
the complex variable; if .U1 ; z1 / is a second open set, then z D u.z1 / and z1 D v.z/ with
u and v holomorphic functions on U \ U1 . To give a differential form ! on C , one has to
give an expression f .z/dz on each .U; z/ such that, on U \ U1 ,

                               f .u.z1 //  u0 .z1 /  dz1 D f1 .z1 /  dz1 :
   21 For a period, Weil’s foundations of algebraic geometry were the only ones available, but they only applied

over a ground field, and so those who wished to work over a ring were forced to device their own foundations.
72                                    CHAPTER I. ABELIAN VARIETIES: GEOMETRY


A differential form is holomorphic if each of the functions f .z/ is holomorphic (rather than
meromorphic). Let ! be a differential on C and let be a path in U \ U1 ; then
                             Z                 Z
                               f .z/ ı dz D       f1 .z1 / ı dz1 :

Thus, it makes sense to integrate ! along any path in C .

T HEOREM 18.1. The set of holomorphic differentials on C forms a g-dimensional vector
space where g is the genus of C .

    We denote this vector space by .C; ˝ 1 /. If !1 ; :::; !g is a basis for the space,R then
every holomorphic differential is a linear combination, ! D ˙ ai !i , of the !i , and ! D
  R                                                                     R            R
a !i ; therefore it suffices to understand the finite set of integrals f !1 ; : : : ; !g g.
    Recall (from topology) that C is a g-holed torus, and that H1 .C; Z/ has a canonical
basis 1 ; :::; 2g — roughly speaking each i goes once round one hole. The vectors
                               0 R
                                          !1
                                             1
                                     j

                        j D @         ::         g
                                             A 2 C ; j D 1; :::; 2g
                               B             C
                                  R     :
                                     j
                                          !g

are called the period vectors.

T HEOREM 18.2. The 2g period vectors are linearly independent over R.

    Thus Cg = Zi is a torus. We shall see that in fact it is an abelian variety (i.e., it has
               P
a Riemann form), and that it is the Jacobian variety.
    FixRa point P0 on C . If P is a second point, and is a path from P0 to P , then
! 7!       ! is linear map .C; ˝ 1 / ! C. Note that if we replace with a different
path   0 from P to P , then 0 differs from      by a loop. If the loop is contractible, then
R        R      0
   ! D 0 !; otherwise the two integrals differ by a sum of periods.

T HEOREM 18.3 (JACOBI INVERSION FORMULA ). Let ` be a linear map .C; ˝ 1 / ! C;
then there exist points P1 ; :::; Pg on C and paths i from P to Pi such that
                                               X Z
                                       `.!/ D          !
                                                       i


for all ! 2   .C; ˝ 1 /.

T HEOREM 18.4 (A BEL ). Let P1 ; :::; Pr and Q1 ; :::; Qr be points on C (not necessarily
distinct). Then there exists a meromorphic function f on C with poles exactly at the Pi
and zeros exactly at the Qi if and only if, for all paths i from P to Pi and all paths i0 from
P to Qi , there exists an element 2 H1 .C; Z/ such that
                               XZ            X Z           Z
                                       !               !D      !
                                                   0
                                     i             i


for all ! 2   .C; ˝ 1 /.
18. ABEL AND JACOBI                                                                           73


    Let   2 H1 .C; Z/; then                       Z
                                           ! 7!       !

is a linear function on the vector space   .C; ˝ 1 /, i.e., an element of    .C; ˝ 1 /_ . Thus we
have a map                         Z
                               7!     W H1 .C; Z/ !       .C; ˝ 1 /_

which (18.2) implies is injective. Set

                                J D      .C; ˝ 1 /_ =H1 .C; Z/:

                              .C; ˝ 1 / identifies J with Cg =
                                                                  P
The choice of a basis for                                              Zj , which is therefore a
complex torus.

T HEOREM 18.5. The intersection product

                                 H1 .C; Z/  H1 .C; Z/ ! Z

is a Riemann form on J . Hence J is an abelian variety.
                                           RQ
    Fix a point P on C . As we noted above, P ! doesn’t make sense, because it depends
                                                                                  RQ
on the choice of a path from P to Q. But two choices differ by a loop, and so ! 7! P !
is well-defined as an element of

                                      .C; ˝ 1 /_ =H1 .C; Z/:

Thus we have a canonical map 'P W C ! J sending P to 0.
   Now consider the map

                 X                  X     Z Pi !
                    ni Pi 7! ! 7!     ni      ! W Div0 .C / ! J , :
                                                  P

    The Jacobi inversion formula shows that this map is surjective (in fact it proves more
than that). Abel’s theorem shows that the kernel of the map is precisely the group of princi-
pal divisors. Therefore, the theorems of Abel and Jacobi show precisely that the above map
defines an isomorphism
                                      Pic0 .C / ! J:
N OTES . For more on these topics, see
    Griffiths, P., Introduction to Algebraic Curves, AMS, 1989. (Treats algebraic curves over C.
Chapter V is on the theorems of Abel and Jacobi.)
    Fulton, W., Algebraic Topology, Springer, 1995, especially Chapter 21.
Chapter II

Abelian Varieties: Arithmetic

1     The Zeta Function of an Abelian Variety
We write Fq for a finite field with q elements, F for an algebraic closure of Fq , and Fq m for
the unique subfield of F with q m elements. Thus the elements of Fq m are the solutions of
   m
c q D c.
     For a variety V over Fq , the Frobenius map V W V ! V is defined to be the identity
map on the underlying topological space of V and is the map f 7! f q on OV . For example,
if V D Pn D Proj.kŒX0 ; :::; Xn /, then V is defined by the homomorphism of rings
                                     q
                          Xi 7! Xi W kŒX0 ; :::; Xn  ! kŒX0 ; :::; Xn 

and induces the map on points
                                                q
                      .x0 W    W xn / 7! .x0 W    W xnq /W Pn .F/ ! Pn .F/:

For any regular map 'W W ! V of varieties over Fq , it is obvious that ' ı W D V ı '.
Therefore, if V ,! Pn is a projective embedding of V , then V induces the map
                                                            q
                                 .x0 W ::: W xn / 7! .x0 W ::: W xnq /

on V .F/. Thus V .Fq / is the set of fixed points of V W V .F/ ! V .F/:
    Let A be an abelian variety over Fq . Then A maps 0 to 0 (because 0 2 V .Fq //, and
so it is an endomorphism of A. Recall that its characteristic polynomial P is a monic
polynomial of degree 2g, g D dim.A/, with coefficients in Z.
                                     Q                           ˇ       ˇ
T HEOREM 1.1. Write P .X / D i .X ai /, and let Nm D ˇA.Fq m /ˇ. Then

  (a) Nm D 2g             aim / for all m  1, and
              Q
                i D1 .1
                                            1
    (b) (Riemann hypothesis) jai j D q 2 .
Hence
                                                    1
                   jNm    q mg j  2g  q m.g       2/   C .22g      2g     1/q m.g 1/ :

P ROOF. We first deduce the inequality from the preceding statements. Take m D 1 in (a)
and expand out to get
                                                2g
                                                X
                  ˇ      ˇ
                  ˇA.Fq /ˇ D a1    a2g              a1    ai 1 ai C1    a2g C   
                                                i D1

                                                    75
76                                   CHAPTER II. ABELIAN VARIETIES: ARITHMETIC


The first term on the right is an integer, and in fact a positive integer because it is P .0/ D
deg./, and (b) shows that it has absolute value q g . Hence it equals q g (actually, it easy
to prove directly that deg./ D q g ). The Riemann hypothesis shows that each term
                                                        1
a1    ai 1 ai C1    a2g has absolute value D q g 2 , and so the sum has absolute value
                1
 2g  q g 2 . There are .2g 2g 1/ terms remaining, and each has absolute value
 q g 1 , whence the inequality. We first prove (a) in the case m D 1. The kernel of

                                        idW A.F/ ! A.F/

is A.Fq /. I claim that the map

                                  .d /0 W Tgt0 .A/ ! Tgt0 .A/

is zero — in fact, that this is true for any variety. In proving it, we can replace A with an
open affine neighbourhood U , and embed U into Am some m in such a way that 0 maps to
the origin 0. The map .d /0 on Tgt0 .U / is the restriction of the map .d /0 on Tgt0 .Am /.
                                                           q           q         q 1
But W An ! An is given by the equations Yi D Xi , and d.Xi / D qXi                  D 0 (in
characteristic p). We now find that

                          d.      id/0 D .d /0    .d.id //0 D     1:

Hence  1 is étale at the origin, and so, by homogeneity, it is étale at every point — each
point in the kernel occurs with multiplicity 1. Therefore,
                                 ˇ       ˇ
                                 ˇA.Fq /ˇ D deg. id/:

But, from the definition of P , we know that

                                     deg.    id/ D P .1/;

                   ai /. When we replace  with  m in the above argument, we find that
           Q
and this is .1
                                   ˇ        ˇ
                                   ˇA.Fq m /ˇ D P m .1/:

Recall (10.20) that a1 ; :::; a2g can be interpreted as the eigenvalues of  acting on T` A.
Clearly  m has eigenvalues a1m ; :::; a2g
                                        m
                                           , and so
                                   Y                         Y
                    P m .X / D        .X aim /, P m .1/ D     .1 aim /

which proves (a) for a general m.                                                             2


     Part (b) follows from the next two lemmas.
L EMMA 1.2. Let  be the Rosati involution on End.A/ ˝ Q defined by a polarization of
         
A; then A ı A D qA .

P ROOF. Let D be the ample divisor on A defining the polarization; thus .a/ D ŒDa          D.
We have to show that
                                   _ ı  ı  D q:
Recall that, on points,  _ is the map

                           ŒD 0  7! Œ  D 0 W Pic0 .A/ ! Pic0 .A/:
1. THE ZETA FUNCTION OF AN ABELIAN VARIETY                                                                   77


Let D 0 be a divisor on A (or, in fact any variety defined over Fq /. If D 0 D div.f / near
.P /, then, by definition,   D 0 D div.f ı / near P . But .P / D P and f ı  D f q
(this was the definition of /, and div.f q / D q div.f /; thus   D 0 D qD. Next observe
that, for any homomorphism ˛W A ! A and any point a on A,

                      ˛ ı ta .x/ D ˛.a C x/ D ˛.a/ C ˛.x/ D t˛.a/ ı ˛.x/:

We can now prove the lemma. For any a 2 A.F/, we have

                            . _ ı  ı /.a/ D   ŒD.a/            D
                                                  D Œ  t.a/
                                                          
                                                               D         D
                                                  D Œ.t.a/ ı / D           D
                                                  D Œ. ı ta / D         D
                                                  D Œta   D       D
                                                  D Œta qD      qD
                                                  D q.a/;

as required.                                                                                                   2


L EMMA 1.3. Let A be an abelian variety over a field k (not necessarily finite). Let ˛ be an
element of End.A/ ˝ Q such that ˛  ı ˛ is an integer r; for any root a of P˛ in C, jaj2 D r.

P ROOF. Note that QŒ˛ is a commutative ring of finite-dimension over Q; it is therefore
an Artin ring. According to (Atiyah and Macdonald 1969, Chapter 8)1 , it has onlyT
                                                                                 finitely
many prime ideals mT 1 ; :::;mn each of which is also maximal, every element of    mi is
nilpotent, and QŒ˛= mi is a product of fields
                             T
                    QŒ˛= mi D K1      Kn ; Ki D QŒ˛=mi :
                   T
We first show that mi D 0, i.e., that QŒ˛ has no nonzero nilpotents. Note that QŒ˛
                                                                                 df
is stable under the action of . Let a ¤ 0 2 QŒ˛. Then b D a  a ¤ 0, because
Tr.a  a/ > 0. As b  D b, Tr.b 2 / D Tr.b   b/ > 0, and so b 2 ¤ 0. Similarly, b 4 ¤ 0,
and so on, which implies b is not nilpotent, and so neither is a. Any automorphism  of
QŒ˛ permutes the maximal ideals mi ; it therefore permutes the factors Ki , i.e., there is a
permutation  of f1; 2; :::; ng and isomorphisms i W Ki ! K .i / such that .a1 ; :::; an / D
.b1 ; :::; bn / with b.i/ D i .ai /. In the case that  D ,  must be the trivial permutation,
for otherwise Tr.a  a/ would not always be positive (consider .a1 ; 0; 0; :::/ if .1/ ¤
1). Hence  preserves the factors of QŒ˛, and is a positive-definite involution on each
of them. The involution  extends by linearity (equivalently by continuity) to a positive-
definite involution of QŒ˛˝R. The above remarks also apply to QŒ˛˝R: it is a product of
fields, and  preserves each factor and is a positive-definite involution on each of them. But
now each factor is isomorphic to R or to C. The field R has no nontrivial automorphisms at
all, and so  must act on a real factor of QŒ˛ ˝ R as the identity map. The field C has only
two automorphisms of finite order: the identity map and complex conjugation. The identity
    1 In fact, it is easy to prove this directly. Let f .X/ be a monic polynomial generating the kernel of QŒX !

QŒ˛. Then QŒX =.f .X // ' QŒ˛, and the maximal ideals of QŒ˛ correspond to the distinct irreducible
factors of f .X /.
78                                      CHAPTER II. ABELIAN VARIETIES: ARITHMETIC


map is not positive-definite, and so  must act on a complex factor as complex conjugation.
We have shown: for any homomorphism W QŒ˛ ! C, .˛  / D .˛/: Thus, for any such
homomorphism, r D .˛   ˛/ D j.˛/j2 , and so every root of the minimum polynomial
                                      1
of ˛ in QŒ˛=Q has absolute value r 2 . Now (10.24) completes the proof.                 2


R EMARK 1.4. We have actually proved the following: QŒ is a product of fields, stable
                                                                                    1
under the involution ; under every map  W QŒ ! C , .  / D ./, and jj D q 2 .

     The zeta function of a variety V over k is defined to be the formal power series

                                               tm
                                                 
                                   P                               ˇ       ˇ
                 Z.V; t / D exp      m1 Nm          ;     Nm D ˇA.Fq m /ˇ :
                                                m

Thus

                                             tm                      tm 2
                                                                      
                            P                       1 P
       Z.V; t / D 1 C            m1 Nm           C        m1 Nm           C    2 QŒŒt:
                                             m      2                m
                                      Q
C OROLLARY 1.5. Let Pr .t / D .1 ai ;r t/, where the ai;r for a fixed r run through the
products ai1 ai2    air , 0 < i1 <    < ir  2g, ai a root of P .t/. Then

                                               P1 .t/    P2g 1 .t/
                          Z.A; t / D
                                         P0 .t/P2 .t/    P2g 2 P2g .t/:

P ROOF. Take the logarithm of each side, and use (1.1a) and the identity (from calculus)

                                log.1    t/ D t C t 2 =2 C t 3 =3 C ::::                      2



R EMARK 1.6. (a) The polynomial Pr .t/ is the characteristic polynomial of  acting on
V r
    T` A.
    (b) Let .V; s/ D Z.V; q s /; then (1.1b) implies that the zeros of .V; s/ lie on the lines
<.s/ D 1=2, 3=2; ::., .2g 1/=2 and the poles on the lines <.s/ D 0; 1; :::; 2g, whence its
name “Riemann hypothesis”.


2     Abelian Varieties over Finite Fields
[A future version of the notes will include a complete proof of the Honda-Tate theorem,
assuming the Shimura-Taniyama theorem (proved in the next section).]
    For a field k, we can consider the following category:
    objects: abelian varieties over kI
    morphisms: Mor.A; B/ D Hom.A; B/ ˝ Q:
    This is called the category of abelian varieties up to isogeny, Isab.k/, over k because
two abelian varieties become isomorphic in Isab.k/ if and only if they are isogenous. It is
Q-linear category (i.e., it is additive and the Hom-sets are vector spaces over Q) and (10.1)
implies that every object in Isab.k/ is a direct sum of a finite number of simple objects.
In order to describe such a category (up to a nonunique equivalence), it suffices to list the
isomorphism classes of simple objects and, for each class, the endomorphism algebra. The
2. ABELIAN VARIETIES OVER FINITE FIELDS                                                   79


theorems of Honda and Tate, which we now explain, allow this to be done in the case
k D Fq .
     For abelian varieties A and B, we use Hom0 .A; B/ to denote Hom.A; B/ ˝ Q — it is
a finite-dimensional Q-vector space.
     Let A be a abelian variety over Fq , and let  D A be the Frobenius endomorphism of
A. Then  commutes with all endomorphisms of A, and so lies in the centre of End0 .A/.
If A is simple, then End0 .A/ is a division algebra. Therefore, in this case, QŒ is a field
(not merely a product of fields). An isogeny A ! B of simple abelian varieties defines
an isomorphism End0 .A/! End0 .B/ carrying A into B , and hence mapping QŒA 
isomorphically onto QŒB :
     Define a Weil q-integer to be an algebraic integer such that, for every embedding
                             1
W QŒ ,! C, jj D q 2 , and let W .q/ be the set of Weil q-integers in C. Say that
two elements  and  0 are conjugate,    0 , if any one of the following (equivalent)
conditions holds:
  (a)  and  0 have the same minimum polynomial over Q;
  (b) there is an isomorphism QŒ ! QŒ 0  carrying  into  0 ;
  (c)  and  0 lie in the same orbit under the action of Gal.Qal =Q/ on W .q/.
   For any simple abelian variety A, the image of A in Qal under any homomorphism
QŒA  ,! Qal is a Weil q-integer, well-defined up to conjugacy (see 1.1). The remark
above, shows that the conjugacy class of A depends only on the isogeny class of A.
T HEOREM 2.1. The map A 7! A defines a bijection

             fsimple abelian varieties=Fq g=(isogeny) ! W .q/=(conjugacy).

P ROOF. The injectivity was proved by Tate and the surjectivity by Honda. We discuss the
proof below.                                                                           2


    To complete the description of Isab.Fq / in terms of Weil q-integers, we have to describe
the division algebra End0 .A/ ˝ Q in terms of A , but before we can do that, we need to
review the classification of division algebras over a number field — see CFT Chapter IV.
    A central simple algebra over a field k is a k-algebra R such that:
  (a) R is finite-dimensional over kI
  (b) k is the centre of R;
  (c) R is a simple ring (i.e., it has no 2-sided ideals except the obvious two).
   If R is also a division algebra, we call it a central division algebra over k.
L EMMA 2.2. If R and S are central simple algebras over k, then so also is R ˝k S .

P ROOF. See CFT IV 2.8.                                                                    2


   For example, if R is a central simple algebra over k, then so also is Mr .R/ D R ˝k
Mr .k/. Here Mr .R/ is the R-algebra of r r matrices with coefficients in R.
P ROPOSITION 2.3 (W EDDERBURN ’ S THEOREM ). Every central simple algebra R over k
is isomorphic to Mr .D/ for some r  1 and central division algebra D over k; moreover r
is uniquely determined by R, and D is uniquely determined up to isomorphism.
80                                 CHAPTER II. ABELIAN VARIETIES: ARITHMETIC


P ROOF. CFT IV 1.15, 1.20.                                                                 2


    The Brauer group Br.k/ of a field is defined as follows. Its elements are the isomor-
phism classes of central division algebras over k. If D and D 0 are two such algebras, then,
according to (2.2, 2.3) D ˝k D 0 is isomorphic Mr .D 00 / for some central division alge-
bra D 00 over k, and we set ŒD  ŒD 0  D ŒD 00 . This is a group — the identity element is
Œk, and ŒD 1 D ŒD opp  where D opp has the same underlying set and addition, but the
multiplication is reversed (if ab D c in D, then ba D c in D opp /.
T HEOREM 2.4. For any local field k, there is a canonical injective homomorphism
                                    invW Br.k/ ,! Q=Z
If k is nonarchimedean, inv is an isomorphism; if k D R, then the image is 21 Z=Z; if
k D C, then Br.k/ D 0.

P ROOF. CFT III 2.1, and IV 4.3.                                                           2


R EMARK 2.5. (a) In fact, Br.k/ D 0 for any algebraically closed field k.
   (b) The nonzero element of Br.R/ is represented by the usual (i.e., Hamilton’s original)
quaternions, H D R C Ri C Rj C Rij:

T HEOREM 2.6. For a number field k, there is an exact sequence
                                     M
                     0 ! Br.k/ !           Br.kv / ! Q=Z ! 0
                                            v

Here the sum is over all primes of k (including the infinite primes),
                                                                 P the first map sends ŒD
to the family .ŒD ˝k kv /v , and the second map sends .av / to v inv.av /.

P ROOF. See CFT VIII 4.2 — it is no easier to prove than the main theorem of abelian class
field theory.                                                                           2


R EMARK 2.7. For a number field k and prime v, write invv .D/ for invkv .D ˝ kv /. The
theorem says that a central division algebra D over k is uniquely determined up to isomor-
phism by its invariants invv .D/; moreover, a family .iv /, iv 2 Q=Z, arises from a central
division algebra over k if and only if it satisfies the following conditions,
  ˘ iv D 0 for all but finitely many v,
  ˘ iv 2 21 Z=Z if v is real and iv D 0 if v is complex, and
    P
  ˘    v iv D 0 (in Q=Z):

     We need one further result.

           p For a central division algebra D over a number field k, the order of ŒD
T HEOREM 2.8.
in Br.k/ is ŒD W k: It is also equal to the least common denominator of the numbers
invv .D/.
                                          L
P ROOF. Since the order of an element of v Q=Z is the least common denominator of
its components, the second statement follows directly from Theorem 2.6. The first follows
from the Grunwald-Wang theorem (CFT VIII 2.4); see Reiner, I., Maximal Orders, 32.19
[and the next version of CFT].                                                          2
2. ABELIAN VARIETIES OVER FINITE FIELDS                                                                81


    We now finally state our theorem.

T HEOREM 2.9. Let A be a simple abelian variety over Fq ; let D D End0 .A/ and let
 2 D be the Frobenius element of A. Then:
  (a) The centre of D is QŒ; therefore, D is a central division algebra over QŒ.
  (b) For a prime v of QŒ, let iv D invv .D/. Then kkv D q iv (here k  kv is the nor-
      malized valuation at the prime v; hence invv .D/ D 1=2 if v is real, and invv .D/ D 0
      if v doesn’t divide p or 1); equivalently,

                                                  ordv ./
                                    invv .D/ D             ŒQŒv W Qp :
                                                  ordv .q/
                                    1
  (c) 2 dim.A/ D ŒD W QŒ 2  ŒQŒ W Q:

P ROOF. This was proved by Tate (Inventiones Math. 1966) who, however, neglected to
publish the proof of (b) (see Waterhouse and Milne, Proc. Symp. Pure Math., AMS XX,
1971).                                                                            2


    The injectivity of the map A 7! ŒA  in (2.1) follows easily from Tate’s theorem:

                  Hom.A; B/ ˝ Q` ' Hom.V` A; V` B/ ,                  D Gal.F=Fq /:

In fact, the canonical generator of Gal.F=Fq / acts on V` A and V` B as A and B respec-
tively, and these action are semisimple (i.e., over some extension of Q` there exist bases of
eigenvectors). It is now an easy exercise in linear algebra to prove that:

                            Hom.V` A; V` B/    D #f.i; j /j ai D bj g
                     Q                        Q
where PA .X / D .X ai /, PB .X/ D .X bj /.
    The surjectivity was proved by Honda. I will only sketch the main idea [for the present].
Obviously, we have to construct over Fq sufficient abelian varieties to exhaust all the con-
jugacy classes of Weil numbers, but we can’t write the equations for a single abelian variety
of dimension > 2 over Fq , so how do we proceed? We construct (special) abelian vari-
eties over C, realize them over number fields, and then reduce their equations modulo p, to
obtain abelian varieties over finite fields.
    Recall (Milne 2006, III 3.17) that, for an elliptic curve A over C, either End0 .A/ D Q
or End0 .A/ D E, a quadratic imaginary number field. The first case is typical; the second
is special. In the second case, A is said to have complex multiplication by E.
    In higher dimensions something similar holds. An algebraic number field E is said to be
a CM-field2 if it is quadratic totally imaginary extension of a totally real field F . Equivalent
definition: there is an involution  ¤ 1 of E such that for every embedding  W E ! C,
complex conjugation acts on E as  1 . An abelian variety A is said to have complex
multiplication3 by the CM-field E if E  End.A/ ˝ Q and
  (a) 2 dim A D ŒEW Q, and
    2 CM = complex multiplication
   3 For more on abelian varieties with complex multiplication, see my notes Complex Multiplication (under

Books on my homepage).
82                                   CHAPTER II. ABELIAN VARIETIES: ARITHMETIC


  (b) for some polarization of A, the Rosati involution on End.A/ ˝ Q stabilizes E, and
      acts on it as .
     Typically, an abelian variety of dimension g over C has End0 .A/ D Q; the opposite
extreme is that A has complex multiplication (of course, now there are many intermediate
possibilities).
     Let A be an abelian variety defined over a nonarchimedean local field k (so k is the field
of fractions of a complete discrete valuation ring R, with maximal ideal m say; let R=m D
k0 /. Embed A into projective space Pn , and let a  kŒX0 ; :::; Xn  be the ideal correspond-
ing to A. Let a0 be the image of a \ RŒX0 ; :::; Xn  in k0 ŒX1 ; :::; Xn  D RŒX0 ; :::=mRŒX0 ; :::.
It defines a variety A0 over k0 . In general A0 may be singular, and it may depend on the
choice of the embedding of A into projective space. However, if the embedding can be
chosen so that A0 is nonsingular, then A0 is independent of all choices, and it is again an
abelian variety. In this case, we say that A has good reduction, and we call A0 the re-
duced variety. When A is an abelian variety over a number field k, then we say A has good
reduction at a finite prime v of k if Akv has good reduction .kv Dcompletion of k at v/.

P ROPOSITION 2.10. Let A be an abelian variety over C with complex multiplication by
E. Then A has a model over some number field k, and, after possibly replacing k with a
larger number field, A will have good reduction at every prime of k.

P ROOF. Omitted.                                                                                  2


    We can construct all abelian varieties over C with complex multiplication by a fixed
CM-field E (up to isogeny) as follows. Let ŒE W Q D 2g; the embeddings E ,! C fall
into g conjugate pairs f';  ı 'g (here  is complex conjugation on C/. A CM-type for E is a
choice of g embeddings ˚ D f'1 ; :::; 'g g of E into C, no two of which differ by complex
conjugation (thus there are 2g different CM-types on E/. Let ˚ also denote the map

                              E ! Cg , x 7! .'1 .x/; :::; 'g .x//;

and define A D Cg =˚.OE /. This is a complex torus, which has a Riemann form, and
hence is an abelian variety. Evidently we can let x 2 OE act on A as ˚.x/, and so A has
complex multiplication by E.
     Thus, starting from a CM-field E and a CM-type ˚, we get an abelian variety A, ini-
tially over C. Proposition 2.10 says that A will be defined over some number field, and
moreover (after possibly replacing the number field by a finite extension) it will reduce to
an abelian variety over some finite field Fq . What is the Weil integer of this abelian variety?
     Given a CM-field E and a CM-type ˚, we can a construct a Weil integer as follows.
Let p be a prime ideal of E lying over p. Then ph is principal for some h, say ph D .a/.
                  df Q
I claim that  D '2˚ '.a2n / is Weil q-integer for some power q of p and that, if n is
taken large enough, it is independent of the choice of the element a generating ph :
     Note first that
                  Y                           Y
          N D         '.a2n /  '.a2n / D                 '.a2n / D .NmE=Q an /2 ;
                    '2˚                           '2Hom.E;C/

which is a positive integer. The ideal

                               .NmE=Q a/ D NmE=Q p D .p/f ;
3. ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION                                             83


where f D f .p=p/ (residue class field degree) — see ANT 4.1. Thus   N D q, where
q D p 2nf .p=p/ . Similarly, one shows that the conjugates of  have this property, so that 
is a Weil q-integer.
     Next note that the unit theorem (ANT 5.9) implies that

                             rank.UE / D g       1 D rank.UF /;

where UE and UF are the groups of units in E and F . Let n D .UE W UF /. A different
generator of ph will be of the form a  u, u 2 UE , and un 2 UFQ      . But f'1 jF; :::; 'g jF g
is the full set of embeddings ofQF into R, and so for any c 2 F , '2˚ 'c D NmF =Q c;
in particular, if c 2 UF , then '2˚ 'c D NmF =Q c is a unit in Z, i.e., it is ˙1. Hence
            2n                   n 2              2
Q
   ' .'.au/ / D   NmF =Q .u / D   .˙1/ D .
     After this miraculous calculation, it will come as no surprise that:

T HEOREM 2.11. Let A be the abelian variety Fq obtained by reduction from an abelian
variety of CM-type .E; ˚ /. Then the Weil q-integer associated to A is that constructed by
the above procedure (up to a root of 1).

P ROOF. This is the main theorem of Shimura and Taniyama, Complex Multiplication of
Abelian Varieties and its Applications to Number Theory, 1961. For a concise modern expo-
sition, see Milne, J., The fundamental theorem of complex multiplication, arXiv:0705.3446.2

    After these observations, it is an exercise in number theory to prove that the map in
(2.1) is surjective. For the details, see (Honda, J. Math. Soc. Japan 20, 1968, 83-95), or,
better, (Tate, Séminaire Bourbaki, 1968/69, Exposé 352, Benjamin, New York).


3    Abelian varieties with complex multiplication
Include a proof of the Shimura-Taniyama theorem, and a sketch of the whole theory.
Chapter III

Jacobian Varieties

This chapter contains a detailed treatment of Jacobian varieties. Sections 2, 5, and 6 prove
the basic properties of Jacobian varieties starting from the definition in 1, while the con-
struction of the Jacobian is carried out in 3 and 4. The remaining sections are largely
independent of one another.


1    Overview and definitions
Overview
Let C be a nonsingular projective curve over a field k. We would like to define an abelian
variety J , called the Jacobian variety of C , such that J.k/ D Pic0 .C / (functorially). Un-
fortunately, this is not always possible: clearly, we would want that J.k sep / D Pic0 .Ck sep /;
but then
                    J.k sep / D J.k/ D Pic0 .Ck sep / , D Gal.k sep =k/;
and it is not always true that Pic0 .Ck sep /   D Pic0 .C /. However, this is true when C.k/ ¤
;.

A SIDE 1.1. Let C be a category. An object X of C defines a contravariant functor

                                hX W C ! Set, T 7! Hom.T; X/:

Moreover X 7! hX defines a functor C ! Fun.C; Set/ (category of contravariant functors
from C to sets). We can think of hX .T / as being the set of “T -points” of X. It is very
easy to show that the functor X 7! hX is fully faithful, i.e., Hom.X; Y / D Hom.hX ; hY /
— this is the Yoneda Lemma (AG 1.39). Thus C can be regarded as a full subcategory of
Fun.C; Set/: X is known (up to a unique isomorphism) once we know the functor it defines,
and every morphism of functors hX ! hY arises from a unique morphism X ! Y . A
contravariant functor F W C ! Set is said to be representable if it is isomorphic to hX for
some object X of C, and X is then said to represent F .

Definition of the Jacobian variety.
For varieties V and T over k, set V .T / D Hom.T; V / D hV .T /. For a nonsingular variety
T,
                           PC0 .T / D Pic0 .C  T /=q  Pic0 .T /


                                                 85
86                                                 CHAPTER III. JACOBIAN VARIETIES


(families of invertible sheaves of degree zero on C parametrized by T , modulo trivial
families–cf. (4.16)). This is a contravariant functor from the category of varieties over
k to the category of abelian groups.

T HEOREM 1.2. Assume C.k/ ¤ ;. The functor PC0 is represented by an abelian variety
J.

     From (1.1), we know that J is uniquely determined. It is called the Jacobian variety of
C.
    A pointed variety over k is a pair .T; t/ with T a variety over k and t 2 T .k/. We
always regard an abelian variety as a pointed variety by taking the distinguished point to
be 0. A divisorial correspondence between two pointed varieties .S; s/ and .T; t/ is an
invertible sheaf L on S  T whose restrictions to S  ftg and fsg  T are both trivial.

P ROPOSITION 1.3. Let P 2 C.k/, and let J D Jac.C /. There is a divisorial correspon-
dence M on C J that is universal in the following sense: for any divisorial correspondence
L on C  T (some pointed variety T / such that Lt is of degree 0 for all t , there is a regular
map 'W T ! J sending the distinguished point of T to 0 and such that .1  '/ M  L.

R EMARK 1.4. (a) The Jacobian variety is defined even when C.k/ D ;; however, it then
doesn’t (quite) represent the functor P (because the functor is not representable). See below.
    (b) The Jacobian variety commutes with extension of scalars, i.e., Jac.Ck 0 / D .Jac.C //k 0
for any field k 0  k.
    (c) Let M be the sheaf in (1.3); as x runs through the elements of J.k/, Mx runs
through a set of representatives for the isomorphism classes of invertible sheaves of degree
0 on C .
    (d) Fix a point P0 in J.k/. There is a regular map 'P0 W C ! J such that, on points,
'P0 sends P to ŒP P0 ; in particular, 'P0 sends P0 to 0. The map 'Q0 differs from 'P0
by translation by ŒP0 Q0  (regarded as a point on J /.
    (e) The dimension of J is the genus of C . If C has genus zero, then Jac.C / D 0 (this
is obvious, because Pic0 .C / D 0, even when one goes to the algebraic closure). If C has
genus 1, then Jac.C / D C (provided C has a rational point; otherwise it differs from C —
because Jac.C / always has a point).

Construction of the Jacobian variety.
Fix a nonsingular projective curve over k. For simplicity, assume k D k al . We want to
construct a variety such that J.k/ is the group of divisor classes of degree zero on C . As a
first step, we construct a variety whose points are the effective divisors of degree r, some
r > 0. Let C r D C  C  :::  C (r copies). A point on C r is an ordered r-tuple of points
on C . The symmetric group on r letters, Sr , acts on C r by permuting the factors, and the
                                      df
points on the quotient variety C .r/ D C r =Sr are the unorderedPr-tuples of points on C .
But an unordered r-tuple is just an effective divisor of degree r, i Pi . Thus
                                   df
                C .r/ D Divr .C / D feffective divisors of degree r on C g:

Write  for the quotient map C r ! C .r/ , .P1 ; :::; Pr / 7!
                                                              P
                                                                Pi .

L EMMA 1.5. The variety C .r/ is nonsingular.
1. OVERVIEW AND DEFINITIONS                                                                    87


P ROOF. In general, when a finite group acts freely on a nonsingular variety, the quotient
will be nonsingular. In our case, there are points on C r whose stabilizer subgroup is non-
trivial, namely the points .P1 ; :::; Pr / in which two (or more) Pi coincide, and we have to
show that they don’t give singularities on the quotient variety. The worst case is a point
Q D .P; :::; P /, and here one can show that
                                     O
                                     bQ ' kŒŒ1 ; :::; r ;

the power series ring in the elementary symmetric functions 1 ; :::; r in the Xi , and this is
a regular ring. See (3.2).                                                                    2


   Let Picr .C / be the set of divisor classes of degree r. For a fixed point P0 on C , the
map
                         ŒD 7! ŒD C rP0 W Pic0 .C / ! Picr .C /
is a bijection (both Pic0 .C / and Picr .C / are fibres of the map degW Pic.C / ! Z). This
remains true when we regard Pic0 .C / and Picr .C / as functors of varieties over k (see
above), and so it suffices to find a variety representing the Picr .C /.
    For a divisor of degree r, the Riemann-Roch theorem says that
                               `.D/ D r C 1       g C `.K       D/
where K is the canonical divisor. Since deg.K/ D 2g 2, deg.K D/ < 0 and `.K D/ D
0 when deg.D/ > 2g 2. Thus,
                     `.D/ D r C 1       g > 0, if r D deg.D/ > 2g           2:
In particular, every divisor class of degree r contains an effective divisor, and so the map
                 'W feffective divisors of degree rg ! Picr .C /, D 7! ŒD
is surjective when r > 2g      2. We can regard this as a morphism of functors
                                     'W C .r/  Picr .C /:
    Suppose that we could find a section s to ', i.e., a morphism of functors sW Picr .C / !
C .r/ such that ' ı s D id. Then s ı ' is a morphism of functors C .r/ ! C .r/ and hence
by (??) a regular map, and we can form the fibre product:
                                      C .r/                J0
                                        ?                  ?
                                 .1;sı'/y
                                        ?                  ?
                                                           y
                                                   
                                  C .r/  C .r/           C .r/ :
Then the map from
                         df
                  J 0 .k/ D f.a; b/ 2 C .r/  C .r/ j a D b;        b D s ı '.a/g
to Picr .C / sending b to '.b/ is an isomorphism. Thus we will have constructed the Jaco-
bian variety; in fact J 0 will be a closed subvariety of C .r/ . Unfortunately, it is not possible
to find such a section: the Riemann-Roch theorem tells us that, for r > 2g 2, each divi-
sor class of degree r is represented by an .r g/-dimensional family of effective divisors,
and there is no nice functorial way of choosing a representative. However, it is possible to
do this “ locally”, and so construct J 0 as a union of varieties, each of which is a closed
subvariety of an open subvariety of C .r/ . For the details, see 4 below.
88                                                 CHAPTER III. JACOBIAN VARIETIES


Definitions and main statements
Recall that for an algebraic space S , Pic.S/ denotes the group H 1 .S; OS / of isomorphism
classes of invertible sheaves on S, and that S 7! Pic.S/ is a functor from the category of
algebraic spaces over k to that of abelian groups.
    Let C be a complete nonsingular curve over k. The degree of a divisor D D ni Pi on C
is ni Œk.Pi /W k. Since every invertible sheaf L on C is of the form L.D/ for some divisor D,
and D is uniquely determined up to linear equivalence, we can define deg.L/ D deg.D/.
Then
                               deg.Ln / D deg.nD/ D n deg.D/;
and the Riemann-Roch theorem says that
                               .C; Ln / D n deg.L/ C 1      g:
    This gives a more canonical description of deg.L/W when .C; Ln / is written as a poly-
nomial in n, deg.L/ is the leading coefficient. We write Pic0 .C / for the group of isomor-
phism classes of invertible sheaves of degree zero on C .
    Let T be a connected algebraic space over k, and let L be an invertible sheaf on C  T .
Then (I 4.2) shows that .Ct ; Lnt /, and therefore deg.Lt /, is independent of t ; moreover,
the constant degree of Lt is invariant under base change relative to maps T 0 ! T . Note
that for a sheaf M on C  T , .q  M/t is isomorphic to OCt and, in particular, has degree
0. Let
               PC0 .T / D fL 2 Pic.C  T / j deg.Lt / D 0 all tg=q  Pic.T /:
Thus PC0 .T / is the group of families of invertible sheaves on C of degree 0 parametrized
by T , modulo the trivial families. Note that PC0 is a functor from algebraic spaces over k
to abelian groups. It is this functor that the Jacobian attempts to represent.
T HEOREM 1.6. There exists an abelian variety J over k and a morphism of functors
W PC0 ! J such that W PC0 .T / ! J.T / is an isomorphism whenever C.T / is nonempty.

   Because C is an algebraic variety, there exists a finite Galois extension k 0 of k such that
C.k 0 / is nonempty. Let G be the Galois group of k 0 over k. Then for every algebraic space
T over k, C.Tk 0 / is nonempty, and so .Tk 0 /W PC0 .Tk 0 / ! J.Tk 0 / is an isomorphism. As
                         df
                  J.T / D Mork .T; J / ' Mork 0 .Tk 0 ; Jk 0 /G D J.Tk 0 /G ,
we see that J represents the functor T 7! PC0 .Tk 0 /G , and this implies that the pair .J; /
is uniquely determined up to a unique isomorphism by the condition in the theorem. The
variety J is called the Jacobian variety of C . Note that for any field k 0  k in which C
has a rational point,  defines an isomorphism Pic0 .C / ! J.k 0 /.
    When C has a k-rational point, the definition takes on a more attractive form. A pointed
k-space is a connected algebraic k-space together with an element s 2 S.k/. Abelian vari-
eties will always be regarded as being pointed by the zero element. A divisorial correspon-
dence between two pointed spaces .S; s/ and .T; t/ over k is an invertible sheaf L on S  T
such that LjS  ftg and Ljfsg  T are both trivial.
T HEOREM 1.7. Let P be a k-rational point on C . Then there is a divisorial correspon-
dence MP between .C; P / and J such that, for every divisorial correspondence L between
.C; P / and a pointed k-scheme .T; t/, there exists a unique morphism 'W T ! J such that
'.t / D 0 and .1  '/ MP  L.
1. OVERVIEW AND DEFINITIONS                                                                 89


   Clearly the pair .J; MP / is uniquely determined up to a unique isomorphism by the
condition in (1.7). Note that each element of Pic0 .C / is represented by exactly one sheaf
Ma , a 2 J.k/, and the map 'W T ! J sends t 2 T .k/ to the unique a such that Ma  Lt .
   Theorem 1.6 will be proved in 4. Here we merely show that it implies (1.7).

L EMMA 1.8. Theorem 1.6 implies Theorem 1.7.

P ROOF. Assume there is a k-rational point P on C . Then for any k -space T , the projection
qW C  T ! T has a section s D .t 7! .P; t//, which induces a map

                     s  D .L 7! LjfP g  T /W Pic.C  T / ! Pic.T /

such that s  q  D id. Consequently, Pic.C  T / D Im.q  / ˚ Ker.s  /, and so PC0 .T / can
be identified with

          P 0 .T / D fL 2 Pic.C  T / j deg.Lt / D 0 all t, LjfP g  T is trivialg.

    Now assume (1.6). As C.T / is nonempty for all k-schemes T , J represents the functor
PC0 D P 0 . This means that there is an element M of P 0 .J / (corresponding to idW J ! J
under / such that, for every k-scheme T and L 2 P 0 .T /, there exists a unique morphism
'W T ! J such that .1  '/ M  L. In particular, for each invertible sheaf L on C
of degree 0, there is a unique a 2 J.k/ such that Ma  L. After replacing M with
.1  ta / M for a suitable a 2 J.k/, we can assume that M0 is trivial, and therefore that
M is a divisorial correspondence between .C; P / and J . It is clear that M has the universal
property required by (1.7).                                                                2


E XERCISE 1.9. Let .J; MP / be a pair having the universal property in (1.7) relative to
some point P on C. Show that J is the Jacobian of C .

   We next make some remarks concerning the relation between PC0 and J in the case that
C does not have a k-rational point.

R EMARK 1.10. For all k-spaces T , .T /W PC0 .T / ! J.T / is injective. The proof of this
is based on two observations. Firstly, because C is a complete variety H 0 .C; OC / D k,
and this holds universally: for any k-scheme T , the canonical map OT ! q OC T is an
                                                                                    
isomorphism. Secondly, for any morphism qW X ! T of schemes such that OT ! q OX ,
the functor M 7! q  M from the category of locally free OT -modules of finite-type to the
category of locally free OX -modules of finite-type is fully faithful, and the essential image
is formed of those modules F on X such that q F is locally free and the canonical map
q  .q F/ ! F is an isomorphism. (The proof is similar to that of I 5.16.)
     Now let L be an invertible sheaf on C  T that has degree 0 on the fibres and which
maps to zero in J.T /; we have to show that L  q  M for some invertible sheaf M on
T . Let k 0 be a finite extension of k such that C has a k 0 -rational point, and let L0 be the
inverse image of L on .C  T /k 0 . Then L0 maps to zero in J.Tk 0 /, and so (by definition
of J / we must have L0  q  M0 for some invertible sheaf M0 on Tk 0 . Therefore q L0 is
locally free of rank one on Tk 0 , and the canonical map q  .q L0 / ! L0 is an isomorphism.
But q L0 is the inverse image of q L under T 0 ! T (see I 4.2a), and elementary descent
theory (cf. 1.13 below) shows that the properties of L0 in the last sentence descend to L;
therefore L  q  M with M D q L.
90                                                  CHAPTER III. JACOBIAN VARIETIES


R EMARK 1.11. It is sometimes possible to compute the cokernel to W PC0 .k/ ! J.k/.
There is always an exact sequence

                               0 ! PC0 .k/ ! J.k/ ! Br.k/

where Br.k/ is the Brauer group of k. When k is a finite extension of Qp , Br.k/ D Q=Z,
and it is known (see Lichtenbaum 1969, p130) that that the image of J.k/ in Br.k/ is
P 1 Z=Z, where P (the period of C / is the greatest common divisor of the degrees of the
k-rational divisors classes on C .

R EMARK 1.12. Regard PC0 as a presheaf on the large étale site over C ; then the pre-
cise relation between J and PC0 is that J represents the sheaf associated with PC0 (see
Grothendieck 1968, 5).

    Finally we show that it suffices to prove (1.6) after an extension of the base field. For
the sake of reference, we first state a result from descent theory. Let k 0 be a finite Galois
extension of a field k with Galois group G, and let V be a variety over k 0 . A descent datum
for V relative to k 0 =k is a collection of isomorphisms ' W V ! V , one for each  2 G,
such that ' D ' ı  ' for all  and  . There is an obvious notion of a morphism of
varieties preserving the descent data. Note that for a variety V over k , Vk 0 has a canonical
descent datum. If V is a variety over k and V 0 D Vk 0 , then a descent datum on an OV 0 -
module M is a family of isomorphisms ' W  M ! M such that '  D ' ' for all 
and .

P ROPOSITION 1.13. Let k 0 =k be a finite Galois extension with Galois group G.
  (a) The map sending a variety V over k to Vk 0 endowed with its canonical descent datum
      defines an equivalence between the category of quasi-projective varieties over k and
      that of quasi-projective varieties over k 0 endowed with a descent datum.
  (b) Let V be a variety over k, and let V 0 D Vk 0 . The map sending an OV -module M to
      M0 D OV 0 ˝ M endowed with its canonical descent datum defines an equivalence
      between the category of coherent OV -modules and that of coherent OV 0 -modules
      endowed with a descent datum. Moreover, if M0 is locally free, then so also is M.

P ROOF. (a) See AG 16.23.
    (b) See Serre 1959, V.20, or Waterhouse 1979, 17. (For the final statement, note that
being locally free is equivalent to being flat, and that V 0 is faithfully flat over V .) 2


P ROPOSITION 1.14. Let k 0 be a finite separable extension of k; if (1.6) is true for Ck 0 , then
it is true for C .

P ROOF. After possibly enlarging k 0 , we may assume that it is Galois over k (with Galois
group G, say) and that C.k 0 / is nonempty. Let J 0 be the Jacobian of Ck 0 . Then J 0 represents
PC0 0 , and so there is a universal M in PC0 .J 0 /. For any  2 G, M 2 PC0 .J 0 /, and so
   k
there is a unique map ' W J 0 ! J 0 such that .1  ' / M D M (in PC0 .J 0 //. One
checks directly that ' D ' ı ' ; in particular, ' ' 1 D 'id , and so the ' are
isomorphisms and define a descent datum on J 0 . We conclude from (1.13) that J 0 has a
2. THE CANONICAL MAPS FROM C TO ITS JACOBIAN VARIETY                                         91


model J over k such that the map PC0 .Tk 0 / ! J.Tk 0 / is G-equivariant for all k-schemes
T . In particular, for all T , there is a map
                                                 
                        PC0 .T / ! PC0 .Tk 0 /G ! J.k 0 /G D J.k/:

To see that the map is an isomorphism when C.T / is nonempty, we have to show that in
this case PC0 .T / ! PC0 .Tk 0 /G is an isomorphism. Let s 2 C.T /; then (cf. the proof of
(1.8)), we can identify PC0 .Tk 0 / with the set of isomorphism classes of pairs .L; ˛/ where
L is an invertible sheaf on C  Tk 0 whose fibres are of degree 0 and ˛ is an isomorphism
        
OTk0 ! .s; 1/ L. Such pairs are rigid—they have no automorphisms—and so each such
pair fixed under G has a canonical descent datum, and therefore arises from an invertible
sheaf on C  T .                                                                            2



2    The canonical maps from C to its Jacobian variety
Throughout this section, C will be a complete nonsingular curve, and J will be its Jacobian
variety (assumed to exist).

P ROPOSITION 2.1. The tangent space to J at 0 is canonically isomorphic to H 1 .C; OC /;
consequently, the dimension of J is equal to the genus of C .

P ROOF. The tangent space T0 .J / is equal to the kernel of J.kŒ"/ ! J.k/, where kŒ" is
ring in which "2 D 0 (AG, Chapter 5). Analogously, we define the tangent space T0 .PC0 /
to PC0 at 0 to be the kernel of PC0 .kŒ"/ ! PC0 .k/. From the definition of J , we obtain a
map of k-linear vector spaces T0 .PC0 / ! T0 .J / which is an isomorphism if C.k/ ¤ ;.
Since the vector spaces and the map commute with base change, it follows that the map is
always an isomorphism.
    Let C" D CkŒ" ; then, by definition, PC0 .kŒ"/ is equal to the group of invertible sheaves
on C" whose restrictions to the closed subscheme C of C" have degree zero. It follows that
T0 .PC0 / is equal to the kernel of H 1 .C" ; O         1      
                                               C" / ! H .C; O C /. The algebraic space C" has
the same underlying topological space as C , but

                            OC" D OC ˝k kŒ" D OC ˚ OC ":

Therefore we can identify the sheaf O                             
                                     C" on C" with the sheaf O C ˚ O C " on C , and so
H 1 .C" ; O       1             1
           C" / D H .C; O C / ˚ H .C; O C "/ . It follows that the map

                           a 7! exp.a"/ D 1 C a"W OC ! O
                                                        C" ;

induces an isomorphism H 1 .C; OC / ! T0 .PC0 /.                                              2


    Let P 2 C.k/, and let LP be the invertible sheaf L. C  fP g fP g  C / on
C  C , where  denotes the diagonal. Note that LP is symmetric and that LP jC  fQg 
L.Q P /. In particular, LP jfP g  C and LP jC  fP g are both trivial, and so LP
is a divisorial correspondence between .C; P / and itself. Therefore, according to (1.7)
there is a unique map f P W C ! J such that f P .P / D 0 and .1  f P / MP  LP .
When J.k/ is identified with Pic0 .C /, f P W C.k/ ! J.k/ becomes identified with the map
Q 7! L.Q/˝L.P / 1 (or, in terms of divisors, the map sending Q to the linear equivalence
92                                                   CHAPTER III. JACOBIAN VARIETIES


class ŒQ P  of Q P ). Note that the map Q nQ Q 7! Q nQ f P .Q/ D Œ Q nQ Q
                                                P                P                     P

from the group of divisors of degree zero on C to J.k/ induced by f P is simply the map
defined by . In particular, it is independent of P , is surjective, and its kernel consists of the
principal divisors.
    From its definition (or from the above descriptions of its action on the points) it is clear
                                             0
that if P 0 is a second point on C , then f P is the composite of f P with the translation map
tŒP P 0  , and that if P is defined over a Galois extension k 0 of k, then f P D f P for all
 2 Gal.k 0 =k/.
    If C has genus zero, then (1.6) shows that J D 0. From now on we assume that C has
genus g > 0.

P ROPOSITION 2.2. The map f  W .J; ˝J1 / !            .C; ˝C1 / is an isomorphism.

P ROOF. As for any group variety, the canonical map hJ W .J; ˝J1 / ! T0 .J /_ is an iso-
morphism Shafarevich 1994, III, 5.2. Also there is a well known duality between .C; ˝C1 /
and H 1 .C; OC /. We leave it as an exercise to the reader (unfortunately rather complicated)
to show that the following diagram commutes:
                                              f
                                 .J; ˝J1 /      !       .C; ˝C1 /
                                    ?                     ?
                                 hJ y'                    y'
                                    ?                     ?

                                               '
                                 T0 .J /_       ! H 1 .C; OC /_
(the bottom isomorphism is the dual of the isomorphism in (1.6)).                                2


P ROPOSITION 2.3. The map f P is a closed immersion (that is, its image f P .C / is closed
and f P is an isomorphism from C onto f P .C //; in particular, f P .C / is nonsingular.

     It suffices to prove this in the case that k is algebraically closed.

L EMMA 2.4. Let f W V ! W be a map of varieties over an algebraically closed field k,
and assume that V is complete. If the map V .k/ ! W .k/ is injective and, for all points
Q of V , the map on tangent spaces TQ .V / ! Tf Q .W / is injective, then f is a closed
immersion.

P ROOF. The image of f is closed because V is complete, and the condition on the tangent
spaces (together with Nakayama’s lemma) shows that the maps Of Q ! OQ on the local
rings are surjective.                                                                 2


P ROOF ( OF 2.3) We apply the lemma to f D f P . If f .Q/ D f .Q0 / for some Q and
Q0 in C.k/, then the divisors Q P and Q0 P are linearly equivalent. This implies that
Q Q0 is linearly equivalent to zero, which is impossible if Q ¤ Q0 because C has genus
> 0. Consequently, f is injective, and it remains to show that the maps on tangent spaces
.df P /Q W TQ .C / ! TfQ .J / are injective. Because f Q differs from f P by a translation,
it suffices to do this in the case that Q D P . The dual of .df P /P W TP .C / ! T0 .J / is
clearly
                                        f               hC
                              .J; ˝ 1 / !     .C; ˝ 1 / ! TP .C /_ ;
2. THE CANONICAL MAPS FROM C TO ITS JACOBIAN VARIETY                                        93


where hC is the canonical map, and it remains to show that hC is surjective. The kernel of
hC is f! 2 .C; ˝ 1 /j!.P / D 0g D .C; ˝ 1 . P //, which is dual to H 1 .C; L.P //. The
Riemann-Roch theorem shows that this last group has dimension g 1, and so Ker.hC / ¤
 .C; ˝ 1 /: hC is surjective, and the proof is complete.                                2


    We now assume that k D C and sketch the relation between the abstract and classical
definitions of the Jacobian. In this case, .C.C/; ˝ 1 / (where ˝ 1 denotes the sheaf of
holomorphic differentials in the sense of complex analysis) is a complex vector space of
dimension   g, and one shows in the theory of abelian integrals that the map  7! .! 7!
                                                                 .C.C/; ˝ 1 /_ . Therefore
R
  !/ embeds H1 .C.C/; Z/ as a lattice into the dual space
    df
J an D   .C.C/; ˝ 1 /_ =H1 .C.C/; Z/ is a complex torus, and the pairing

                            H1 .C.C/; Z/  H1 .C .Z/; Z/ ! Z

defined by Poincaré duality gives a nondegenerate Riemann form on J an . Therefore J an
is an abelian variety over C. For each P there isR a canonical map g P W C ! J an sending
a point Q to the element represented by .! 7!       !/, where is any path from P to Q.
Define eW .C.C/; ˝ 1 /_ ! J .C/ to be the surjection in the diagram:
                                                    onto
                                .C.C/; ˝ 1 /_          ! J.C/
                                    ?                     x
                                   'yf _
                                    ?                     ?exp
                                                          ?
                                                    '
                                   .J; ˝ 1 /_          ! T0 .J /:

Note that if .C.C/; ˝ 1 /_ is identified with T                          P
                                              R P .C /, then .de/0 DP .df /P . It follows
that if is a path from P to Q and ` D .! 7!       !/, then e.`/ D f .Q/.

T HEOREM 2.5. The canonical surjection eW .C.C/; ˝ 1 /_  J.C / induces an isomor-
phism J an ! J carrying g P into f P .

P ROOF. We have to show that the kernel of e is H1 .C.C/; Z/, but this follows from Abel’s
theorem and the Jacobi inversion theorem.
    (Abel): Let P1 ; :::; Pr and Q1 ; :::; Qr be elements of C.C/; then there is a meromorphic
function on C.C/ with its poles at the Pi and its zeros at the Qi if and only if for any paths
                         0
 i from P to Pi and i from P to Qi there exists a         in H1 .C.C/; Z/ such that
                              XZ            XZ          Z
                                      !             !D      ! all !.
                                                0
                                   i            i


     (Jacobi) Let ` be a linear mapping .C.C/; ˝ 1 / ! C. Then therePexist        R g points
P1 ; :::; Pg on C.C/ and paths 1 ; :::; g from P to Pi such that `.!/ D         i   i
                                                                                      ! for all
! 2 .C.C/; ˝ /.  1

     Let ` 2 .C.C/; ˝ 1 /_ ; we may assume P   it is defined by g points P1 ; :::; Pg . Then `
maps to zero in J.C/ if and only if the divisor Pi gP is linearly equivalent to zero, and
Abel’s theorem shows that this is equivalent to ` lying in H1 .C.C/; Z/.                     2
94                                                          CHAPTER III. JACOBIAN VARIETIES


3     The symmetric powers of a curve
Both in order to understand the structure of the Jacobian, and as an aid in its construction,
we shall need to study the symmetric powers of C .
     For any variety V , the symmetric group Sr on r letters acts on the product of r copies
V of V by permuting the factors, and we want to define the rth symmetric power V .r/ of
   r

V to be the quotient Sr nV r . The next proposition demonstrates the existence of V .r/ and
lists its main properties.
     A morphism 'W V r ! T is said to be symmetric if ' D ' for all  in Sr .
P ROPOSITION 3.1. Let V be a variety over k. Then there exists a variety V .r/ and a
symmetric morphism W V r ! V .r/ having the following properties:
    (a) as a topological space, .V .r/ ; / is the quotient of V r by Sr I
    (b) for any open affine subset U of V , U .r/ is an open affine subset of V .r/ and

                                       .U .r/ ; OV .r/ / D      .U r ; OV r /Sr

        (set of elements fixed by the action of Sr /.
The pair .V .r/ ; / has the following universal property: every symmetric k-morphism
'W V r ! T factors uniquely through .
    The map  is finite, surjective, and separable.

P ROOF. If V is affine, say V D Specm.A/,          .r/ to be Specm..A ˝ ::: ˝ A/Sr /.
                                       S define V                          k      k
In the general case, write V as a union Ui of open affines, and construct V by patching
               .r/
together the Ui . See Mumford 1970, II, 7, p66, and III, 11, p112, for the details. 2


   The pair .V .r/ ; / is uniquely determined up to a unique isomorphism by its universal
property. It is called the rth symmetric power of V .
P ROPOSITION 3.2. The symmetric power C .r/ of a nonsingular curve is nonsingular.

P ROOF. We may assume that k is algebraically closed. The most likely candidate for a
singular point on C .r/ is the image Q of a fixed point .P; :::; P / of Sr on C r , where P is
a closed point of C. The completion OO P of the local ring at P is isomorphic to kŒŒX, and
so
                     OO .P;:::;P /  kŒŒX Ő ::: Ő kŒŒX  kŒŒX1 ; :::; Xr .
It follows that OO Q  kŒŒX1 ; :::; Xr Sr where Sr acts by permuting the variables. The fun-
damental theorem on symmetric functions says that, over any ring, a symmetric polynomial
can be expressed as a polynomial in the elementary symmetric functions 1 ; :::; r . This
implies that
                             kŒŒX1 ; :::; Xr Sr D kŒŒ1 ; :::; r ,
which is regular, and so Q is nonsingular.
    For a general point Q D .P; P; :::; P 0 ; :::/ with P occurring r 0 times, P 0 occurring
 00
r times, and so on,

                     OO Q  kŒŒX1 ; :::; Xr 0 Sr 0 Ő kŒŒX1 ; :::; Xr 00 Sr 00 Ő ...,

which the same argument shows to be regular.                                                 2
3. THE SYMMETRIC POWERS OF A CURVE                                                             95


R EMARK 3.3. The reader may find it surprising that the fixed points of the action of Sr on
C r do not force singularities on C .r/ . The following remarks may help clarify the situation.
Let G be a finite group acting effectively on a nonsingular variety V , and supppose that the
quotient variety W D GnV exists. Then V ! W is ramified exactly at the fixed points of
the action. A purity theorem Grothendieck 1971, X, 3.1, says W can be nonsingular only if
the ramification locus is empty or has pure codimension 1 in V . As the ramification locus
of V r over V .r/ has pure codimension dim.V /, this implies that V .r/ can be nonsingular
only if V is a curve.

    Let K be field containing k. If K is algebraically closed, then (3.1a) shows that
C .r/ .K/   D Sr nC.K/r , and so a point of C .r/ with coordinates in K is an unordered
r-tuple of K-rational points. This is the same thing as an effective divisor of degree r on
CK . When K is perfect, the divisors on CK can be identified with those on CK fixed un-
der the action of Gal.K al =K/. Since the same is true of the points on C .r/ , we see again
that C .r/ .K/ can be identified with the set of effective divisors of degree r on C. In the
remainder of this section we shall show that C .r/ .T / has a similar interpretation for any k-
scheme. (Since this is mainly needed for the construction of J , the reader more interested
in the properties of J can pass to the section 5.)
    Let X be an algebraic space over k. Recall Hartshorne 1977, II 6 p145, that a Cartier
divisor D is effective if it can be represented by a family .Ui ; gi /i with the gi in .Ui ; OX /.
Let I.D/ be the subsheaf of OX such that I.D/jUi is generated by gi . Then I.D/ D
L. D/, and there is an exact sequence

                                0 ! I.D/ ! OX ! OD ! 0

where OD is the structure sheaf of the closed algebraic subspace of T associated with D.
The closed subspaces arising from effective Cartier divisors are precisely those whose sheaf
of ideals can be locally generated by a single element that is not a zero-divisor. We shall
often identify D with its associated closed subscheme.
    For example, let T D A1 D Specm kŒY , and let D be the Cartier divisor associated
with the Weil divisor nP , where P is the origin. Then D is represented by .Y n ; A1 /, and
the associated algebraic subspace is Specm.kŒY =.Y n //.
D EFINITION 3.4. Let W X ! T be a morphism of k-schemes. A relative effective Cartier
divisor on X=T is a Cartier divisor on X that is flat over T when regarded as an subspace
of X .

    Loosely speaking, the flatness condition means that the divisor has no vertical compo-
nents, that is, no components contained in a fibre. When T is affine, say T D Specm.R/,
an algebraic subspace D of X Sis a relative effective Cartier divisor if and only if there exists
an open affine covering X D Ui and gi 2 .Ui ; OX / D Ri such that
  (a) D \ Ui D Specm.Ri =gi Ri /,
  (b) gi is not a zero-divisor, and
  (c) Ri =gi Ri is flat over R, for all i .
    Henceforth all divisors will be Cartier divisors.
L EMMA 3.5. If D1 and D2 are relative effective divisors on X=T , then so also is their sum
D1 C D2 .
96                                                   CHAPTER III. JACOBIAN VARIETIES


P ROOF. It suffices to prove this in the case that T is affine, say T D Specm.R/. We have
to check that if conditions (b) and (c) above hold for gi and gi0 , then they also hold for
gi gi0 . Condition (b) is obvious, and the flatness of Ri =gi gi0 Ri over R follows from the
exact sequence
                                      gi0
                     0 ! Ri =gi Ri ! Ri =gi gi0 Ri ! Ri =gi0 Ri ! 0;
which exhibits it as an extension of flat modules.                                            2


R EMARK 3.6. Let D be a relative effective divisor on X=T . On tensoring the inclusion
I.D/ ,! OX with L.D/ we obtain an inclusion OX ,! L.D/ and hence a canonical
global section sD of L.D/. For example, in the case that T is affine and D is represented
as in the above example, L.D/jUi is gi 1 Ri and sD jUi is the identity element in Ri .
    The map D 7! .L.D/; sD / defines a one-to-one correspondence between relative ef-
fective divisors on X=T and isomorphism classes of pairs .L; s/ where L is an invertible
sheaf on X and s 2 .X; L/ is such that
                                             s
                              0 ! OX ! L ! L=sOX ! 0

is exact and L=sOX is flat over T .
     Observe that, in the case that X is flat over T , L=sOX is flat over T if and only if, for
all t in T , s does not become a zero divisor in L ˝ OXt . (Use that an R-module M is flat
if TorR1 .M; N / D 0 for all finitely generated modules N , and that any such module N has
a composition series whose quotients are the quotient of R by a prime ideal; therefore the
criterion has only to be checked with N equal to such a module.)

P ROPOSITION 3.7. Consider the Cartesian square

                                            X        X0
                                            ?        ?
                                            ?        ?
                                            y        y
                                            X        T0
If D is a relative effective divisor on X=T , then its pull-back to a closed subspace D 0 of X 0
is a relative effective divisor on X 0 =T 0 .

P ROOF. We may assume both T and T 0 are affine, say T D Specm R and T 0 D Specm R0 ,
and then have to check that the conditions (a), (b), and (c) above are stable under the base
change R ! R0 . Write Ui0 D U T T 0 ; clearly D 0 \ Ui0 D Specm.Ri0 =gi Ri0 /. The
conditions (b) and (c) state that
                                        gi
                             0 ! Ri ! Ri ! Ri =gi Ri ! 0

is exact and that Ri =gi Ri is flat over R. Both assertions continue to hold after the sequence
has been tensored with R0 .                                                                   2


P ROPOSITION 3.8. Let D be a closed subscheme of X, and assume that D and X are both
                     df
flat over T . If Dt D D T ft g is an effective divisor on Xt =t for all points t of T , then D
is a relative effective divisor on X.
3. THE SYMMETRIC POWERS OF A CURVE                                                               97


P ROOF. From the exact sequence

                                0 ! I.D/ ! OX ! OD ! 0

and the flatness of X and D over T , we see that I.D/ is flat over T . The flatness of OD
implies that, for any t 2 T , the sequence

                         0 ! I.D/ ˝OT k.t/ ! OXt ! ODt ! 0
                                           
is exact. In particular, I.D/ ˝ k.t / ! I.Dt /. As Dt is a Cartier divisor, I.Dt / (and
therefore also I.D/ ˝ k.t // is an invertible OXt -module. We now apply the fibre-by-fibre
criterion of flatness: if X is flat over T and F is a coherent OX -module that is flat over
T and such that Ft is a flat OXt -module for all t in T , then F is flat over X (Bourbaki
1989, III, 5.4). This implies that I.D/ is a flat OX -module, and since it is also coherent,
                                                                  
it is locally free over OX . Now the isomorphism I.D/ ˝ k.t/ ! I.Dt / shows that it is
of rank one. It is therefore locally generated by a single element, and the element is not a
zero-divisor; this shows that D is a relative effective divisor.                          2


    Let W C ! T be a proper smooth morphism with fibres of dimension one. If D is a
relative effective divisor on C=T , then Dt is an effective divisor on Ct , and if T is connected,
then the degree of Dt is constant; it is called the degree of D. Note that deg.D/ D r if and
only if OD is a locally free OT -module of degree r.

C OROLLARY 3.9. A closed subspace D of C is a relative effective divisor on C=T if and
only if it is finite and flat over T ; in particular, if sW T ! C is a section to , then s.T / is a
relative effective divisor of degree 1 on C =T .

P ROOF. A closed subspace of a curve over a field is an effective divisor if and only if it is
finite. Therefore (3.8) shows that a closed subspace D of C is a relative effective divisor on
C=T if and only if it is flat over T and has finite fibres, but such a subspace D is proper over
T and therefore has finite fibres if and only if it is finite over T (see Milne 1980, I 1.10, or
Hartshorne 1977, III, Ex. 11.3).                                                              2


   When D and D 0 are relative effective divisors on C=T , we write D  D 0 if D  D 0 as
subspaces of C (that is, I.D/  I.D 0 //.

P ROPOSITION 3.10. If Dt  Dt0 (as divisors on Ct / for all t in T , then D  D 0 .

P ROOF. Represent D as a pair .s; L/ (see 3.6). Then D  D 0 if and only if s becomes zero
in L ˝ OD 0 D LjD 0 . But L ˝ OD 0 is a locally free OT -module of finite rank, and so the
support of s is closed subspace of T . The hypothesis implies that this subspace is the whole
of T .                                                                                     2


    Let D be Sa relative effective divisor of degree r on C=T . We shall say thatP
                                                                                 D is split if
Supp.D/ D si .T / for some sections si to . For example, a divisor D D            ni Pi on a
curve over a field is split if and only if k.Pi / D k for all i .

                P Every split relative effective divisor D on C=T can be written uniquely
P ROPOSITION 3.11.
in the form D D    ni si .T / for some sections si .
98                                                           CHAPTER III. JACOBIAN VARIETIES

                             S
 ROOF. Let Supp.D/ D
PP                                                          P i .T / has degree ni . Then Dt D
                                i si .T /, and suppose that Djs
. ni si .T //t for all t , and so (3.10) shows that D D        ni si .T /.                   2


E XAMPLE 3.12. Consider a complete nonsingular curve C over a field k. For each i there
is a canonical section si to qW C  C r ! C r , namely, .P1 ; :::; Pr / 7! .Pi ; P1 ; :::; PrP
                                                                                             /. Let
Di to be si .C r / regarded as a relative effective divisor on C  C r =C r , and let D D Di .
                                                           r
 P D is the unique relative effective divisor C  C =T whose fibre over .P1 ; :::; Pr / is
Then
    Pi . Clearly D is stable under the action of the symmetric group Sr , and Dcan D Sr nD
(quotient as a subscheme of C  C r ) is a relative effective divisor on C  C .r/ =C .r/ whose
fibre over D 2 C .r/ .k/ is D.

    For C a complete smooth curve over k and T a k-scheme, define DivrC .T / to be the set
of relative effective Cartier divisors on C  T =T of degree r. Proposition 3.7 shows that
DivrC is a functor on the category of k-schemes.
T HEOREM 3.13. For any relative effective divisor D on .C  T / =T of degree r, there is a
unique morphism 'W T ! C .r/ such that D D .1  '/ 1 .Dcan /. Therefore C .r/ represents
DivrC .
                                                         P
P ROOF. Assume first that D is split, so that D D           ni si .T / for some sections si W T !
                                            r
C  T . In this case, we define T ! C to be the map .p ı s1 ; :::; p ı s1 ; p ı s2 ; :::/, where
each si occurs ni times, and we take ' to be the composite T ! C r ! C .r/ . In general,
we can choose a finite flat covering W T 0 ! T such that the inverse image D 0 of D on
C  T 0 is split, and let ' 0 W T 0 ! C .r/ be the map defined by D 0 . Then the two maps ' 0 ı p
and ' 0 ı q from T 0 T T 0 to T 0 are equal because they both correspond to the same relative
effective divisor

                     p 1 .D 0 / D .      ı p/ 1 .D/ D .         ı q/ 1 .D/ D q 1 .D/

on T 0 T T 0 . Now descent theory (Milne 1980, I, 2.17) shows that ' 0 factors through T . 2

E XERCISE 3.14. Let E be an effective Cartier divisor of degree r on C , and define a
subfunctor DivE       r
              C of DivC by

                         DivE                r
                            C .T / D fD 2 DivC .T / j Dt  E all t 2 T g:

Show that DivE  C is representable by P.V / where V is the vector space   .C; L.E// (use
                                                    E       r
Hartshorne 1977, II 7.12, and that the inclusion DivC ,! DivC defines a closed immersion
P.V / ,! C .r/ ).

R EMARK 3.15. Theorem 3.13 says that C .r/ is the Hilbert scheme HilbP
                                                                     C =k
                                                                          where P is
the constant polynomial r.


4     The construction of the Jacobian variety
1 In this section, C will be a complete nonsingular curve of genus g > 0, and P will be a

k-rational point on C. Recall (1.14), that in constructing J , we are allowed to make a finite
separable extension of k.
     1 The method of construction of the Jacobian variety in this section was suggested to me by János Kollár.
4. THE CONSTRUCTION OF THE JACOBIAN VARIETY                                                99


    For an algebraic k-space T , let

                   PCr .T / D fL 2 Pic.C  T / j deg.Lt / D r all tg= ;

where L  L0 means L  L0 ˝ q  M for some invertible sheaf M on T. Let Lr D L.rP /;
then L 7! L˝p  Lr is an isomorphism PC0 .T / ! PCr .T /, and so, to prove (1.6), it suffices
to show that PCr is representable for some r. We shall do this for a fixed r > 2g.
    Note that there is a natural transformation of functors f W DivrC ! PCr sending a relative
effective divisor D on C  T =T to the class of L.D/ (or, in other terms, .s; L/ to the class
of L/.

L EMMA 4.1. Suppose there exists a section s to f W DivrC ! PCr . Then PCr is repre-
sentable by a closed subscheme of C .r/ .

P ROOF. The composite ' D s ı f is a natural transformation of functors DivrC ! DivrC
and DivrC is representable by C .r/ , and so ' is represented by a morphism of varieties.
Define J 0 to be the fibre product,

                                      C .r/              J0
                                        ?                ?
                                   .1;'/y
                                        ?                ?
                                                         y
                                                  
                                  C .r/  C .r/         C .r/
Then

                 J 0 .T / D f.a; b/ 2 C .r/ .T /  C .r/ .T / j a D b, a D 'bg
                        D fa 2 C .r/ .T / j a D '.a/g
                        D fa 2 C .r/ .T / j a D sc, some c 2 PCr .T /g
                         PCr .T /,

because s is injective. This shows that PCr is represented by J 0 , which is a closed subspace
of C .r/ because  is a closed immersion.                                                    2


    The problem is therefore to define a section s or, in other words, to find a natural way
of associating with a family of invertible sheaves L of degree r a relative effective divisor.
For L an invertible sheaf of degree r on C , the dimension h0 .L/ of H 0 .C; L/ is r C 1 g,
and so there is an r g dimensional system of effective divisors D such that L.D/  L.
One way to cut down the size of this system is to fix a family D .P1 ; :::; Pr g / of k-
        points on C and consider only divisors D in the system such that D  D , where
rationalP
D D        Pi . As we shall see, this provides a partial solution to the problem.

P ROPOSITION
   P         4.2. Let        be an .r    g/-tuple of k-rational points on C , and let L    D
L. P 2 P /.

  (a) There is an open subvariety C of C .r/ such that, for all k-schemes T;

                    C .T / D fD 2 DivrC .T / j h0 .Dt      D / D 1, all t 2 T g:

       If k is separably closed, then C .r/ is the union of the subvarieties C .
100                                                 CHAPTER III. JACOBIAN VARIETIES


  (b) For all k-schemes T , define

                     P .T / D fL 2 PCr .T / j h0 .Lt ˝ L 1 / D 1, all t 2 T g:

      Then P is a subfunctor of P and the obvious natural transformation f W C ! P
      has a section.

P ROOF. (a) Note that for any effective divisor D of degree r on C , h0 .D D /  1,
and that equality holds for at least one D (for example, D D D C Q1 C    C Qg for
a suitable choice of points Q1 ; :::; Qg ; see the elementary result (5.2b) below). Let Dcan
be the canonical relative effective divisor of degree r on C  C .r/ =C .r/ . Then (I 4.2c)
applied to L.Dcan p 1 D / shows that there is an open subscheme C of C .r/ such that
h0 ..Dcan /t D / D 1 for t in C and h0 ..Dcan /t D / > 1 otherwise. Let T be an
algebraic k-space, and let D be a relative effective divisor of degree r on C  T =T such
that h0 .Dt D / D 1. Then (3.13) shows that there is a unique morphism 'W T ! C .r/
such that .1  '/ 1 .Dcan / D D, and it is clear that ' maps T into C . This proves the first
assertion.                                                      S
    Assume S  that k is separably closed.  To  show    that C D    C , it suffices to show that
C.k/ D        C .k/, or that for every divisor D of degree r on C , there exists a such
that h0 .D D / D 1. Choose a basis e0 ; :::; er g for H 0 .C; L.D//, and consider the
                                       r g . Then .C / is not contained in any hyperplane (if
corresponding embedding P W C ,! P          P
it were contained in      ai Xi D 0, then        ai ei would be zero on C ), and so there exist
r g points P1 ; :::; Pr g on C disjoint from D whose images are not contained in linear
subspace of codimension 2 (choose P1 ; P2 ; :.. inductively so that P1 ; :::; Pi is not contained
in a linear subspace of dimension i 2). The .r g/-tuple D .P1 ; :::; Pr g / satisfies the
condition because

         H 0 .C; L.D
                          P           ˚P          P
                             Pj // D      ai ei j ai ei .Pj / D 0, j D 1; :::; r g ;

which has dimension < 2.
    (b). Let L be an invertible sheaf on C  T representing an element of P .T /. Then
h0 .Dt D / D 1 for all t, and the Reimann-Roch theorem shows that h1 .Dt D / D 0
                                       df
for all t. Now I, 4.2e, shows that M D q .L ˝ p  L 1 / is an invertible sheaf on T and that
its formation commutes with base change. This proves that PC is a subfunctor of PCr . On
tensoring the canonical map q  M ! L ˝ p  L 1 with q  M 1 , we obtain a canonical
map OC T ! L ˝ p  L 1 ˝ q  M 1 . The natural map L ! OC induces a map
p  L 1 ! OC T , and on combining this with the preceding map, we obtain a canonical
map s W OC T ! L ˝ q  M 1 . The pair .s ; L ˝ q  M 1 / is a relative effective divisor
on C  T =T whose image under f in P .T / is represented by L ˝ q  M 1  L. We have
defined a section to C .T / ! P .T /, and our construction is obviously functorial.        2


C OROLLARY 4.3. The functor P is representable by a closed subvariety J of C .

P ROOF. The proof is the same as that of (4.1).                                                2

                                                                                                0
P ROOF ( OF T HEOREM 1.6) Now consider two .g r/-tuples and 0 , and define P ;
                                0                 0
to be the functor such that P ; .T / D P .T / \ P .T / for all k-schemes T . It easy to see
5. CANONICAL MAPS                                                                                 101

          0                                    0                          0                   0
that P ; is representable by a variety J ; such that the maps J ; ,! J and J ; ,!
   0                                0                    0      0
J defined by the inclusions P ; ,! P and P ; ,! P are open immersions.
                                 S the Jacobian of C . Choose tuples 1 ; :::; m of points
     We are now ready to construct
       sep               .r/
in C.k / such that C         D      C i . After extending k, we can assume that the i are
tuples of k-rational points. Define J by patching together the varieties J i using the open
immersions J i ; j ,! J i ; J j . It is easy to see that J represents the functor PCr , and
therefore also the functor PC0 . Since the latter is a group functor, J is a group variety. The
natural transformations DivrC ! PCr ! PC0 induce a morphism C .r/ ! J , which shows
that J is complete and is therefore an abelian variety.                                       2



5    The canonical maps from the symmetric powers of C to its
     Jacobian variety
Throughout this section C will be a complete nonsingular curve of genus g > 0. Assume
there is a k-rational point P on C , and write f for the map f P defined in 2.
    Let f r be the map C r ! J sending .P1 ; :::; Pr / to f .P1 / C    C f .Pr /. On points,
  r
f is the map .P1 ; :::; Pr / 7! ŒP1 C    C Pr rP . Clearly it is symmetric, and so
induces a map f .r/ W C .r/ ! J . We can regard f .r/ as being the map sending an effective
divisor D of degree r on C to the linear equivalence class of D rP . The fibre of the map
f .r/ W C .r/ .k/ ! J.k/ containing D can be identified with the space of effective divisors
linearly equivalent to D, that is, with the linear system jDj. The image of C .r/ in J is
a closed subvariety W r of J , which can also be written W r D f .C / C    C f .C / (r
summands).
T HEOREM 5.1. (a) For all r  g, the morphism f .r/ W C .r/ ! W r is birational; in partic-
ular, f .g/ is a birational map from C .g/ onto J .
     (b) Let D be an effective divisor of degree r on C , and let F be the fibre of f .r/
containing D. Then no tangent vector to C .r/ at D maps to zero under .df .r/ /D unless it
lies in the direction of F ; in other words, the sequence
                   0 ! TD .F / ! TD .C .r/ / ! Ta .J /;        a D f .r/ .D/;
is exact. In particular, .df .r/ /D W TD .C .r/ / ! Ta .J / is injective if jDj has dimension zero.

    The proof will occupy the rest of this section.
    For D a divisor on C , we write h0 .D/ for the dimension of
                         H 0 .C; L.D// D ff 2 k.C /j.f / C D  0g
and h1 .D/ for the dimension of H 1 .C; L.D//. Recall that
                             h0 .D/    h1 .D/ D deg.D/ C 1         g;
and that H 1 .C; L.D//_ D H 0 .C; ˝ 1 . D//, which can be identified with the set of ! 2
  1
˝k.C  /=k
          whose divisor .!/  D.

L EMMA 5.2. (a) Let D be a divisor on C such that h1 .D/ > 0; then there is a nonempty
open subset U of C such that h1 .D C Q/ D h1 .D/ 1 for all points Q in U , and
h1 .D C Q/ D h1 .D/ for Q … U .
    (b) For any r  g, there is an open subset U of C r such that h0 . Pi / D 1 for all
                                                                      P
.P1 ; :::; Pr / in U .
102                                                  CHAPTER III. JACOBIAN VARIETIES


P ROOF. (a) If Q is not in the support of D, then

                       H 1 .C; L.D C Q//_ D          .C; ˝ 1 . D        Q//

can be identified with the subspace of .C; ˝ 1 . D// of differentials with a zero at Q.
Clearly therefore we can take U to be the complement of the zero set of a basis of H 1 .C; L.D//
together with a subset of the support of D.
    (b) Let D0 be the divisor zero on C . Then h1 .D0 / D Pg, and on applying (a) repeatedly,
                                           r
we find that there is an open subset U of C such that hP1 . Pi / D g r for all .P1 ; :::; Pr /
in U . The Riemann-Roch theorem now shows that h0 . Pi / D r C .1 g/ C .g r/ D 1
for all .P1 ; :::; Pr / in U .                                                              2


    In proving (5.1), we can assume that k is algebraically closed. If U 0 is the image in
C .r/ of the set U in (5.2b), then f .r/ W C .r/ .k/ ! J.k/ is injective on U 0 .k/, and so
f .r/ W C .r/ ! W r must either be birational or else purely inseparable of degree > 1. The
second possibility is excluded by part (b) of the theorem, but before we can prove that we
need another proposition.
P ROPOSITION 5.3. (a) For all r  1, there are canonical isomorphisms
                                  '                      '
                       .C; ˝ 1 / !      .C r ; ˝ 1 /Sr !        .C .r/ ; ˝ 1 /:

Let ! 2 .C; ˝ 1 / correspond to ! 0 2 .C .r/ ; ˝ 1 /; then for any effective divisor D of
degree r on C , .!/  D if and only if ! 0 has a zero at D.
    (b) For all r  1, the map f .r/ W .J; ˝ 1 / ! .C .r/ ; ˝ 1 / is an isomorphism.

    A global 1-form on a product of Lprojective   varieties is a sum of global 1-forms on the
                          r   1           
factors. Therefore .C ; ˝ / D i pi .C; ˝ 1 /, where        P the    pi are the projection maps
onto the factors, and so it is clear that the map ! 7!         pi ! identifies .C; ˝ 1 / with
  .C r ; ˝ 1 /Sr . Because W C r ! C .r/ is separable,   W .C .r/ ; ˝ 1 / ! .C r ; ˝ 1 / is
injective, and its image is obviously fixed by the action of Sr . The composite map

                  .J; ˝ 1 / !   .C .r/ ; ˝ 1 / ,!  .C r ; ˝ 1 /Sr ' .C; ˝ 1 /
                                                                P  0
sends ! to the element ! 0 of .C; ˝ 1 / such that f r ! D        pi ! . As f r D
                                                                                   P
                                                                                      f ı pi ,
clearly ! 0 D f  !, and so the composite map is f  which we know to be an isomorphism
(2.2). This proves that both maps in the above sequence are isomorphisms. It also com-
pletes the proof of the proposition except for the second part of (a), and for this we need a
combinatorial lemma.
L EMMA 5.4.
         P Let 1 ; :::; r be the elementary symmetric polynomials in X1 ; :::; Xr , and
            j
let j D  Xi dXi . Then

               m 0   m 1 1 C    C . 1/m m D dmC1 , all m  r             1:

P ROOF. Let m .i / be the mth elementary symmetric polynomial in the variables

                                X1 ; :::; Xi 1 ; Xi C1 ; :::; Xr :

Then
                            m n D m n .i/ C Xi m n 1 .i/;
5. CANONICAL MAPS                                                                           103


and on multiplying this by . 1/n Xin and summing over n (so that the successive terms
cancel out) we obtain the identity

                        m     m 1 Xi C    C . 1/m Xim D m .i/:

On multiplying this with dXi and summing, we get the required identity.                       2


     We now complete the proof of (5.3). First let D D rQ. Then OO Q D kŒŒX and
OO D D kŒŒ1 ; :::; r  (see the proof of (3.2); by OD we mean the local ring at the point D
on C .r/ ). If ! D .a0 C a1 X C a2 X 2 C    /dX , ai 2 k, when regarded as an element of
˝ 1O    , then ! 0 D a0 0 C a1 1 C    . We know that fd1 ; :::; dr g is a basis for ˝ 1O
  OQ =k                                                                                   OQ =k
as an OO D -module, but the lemma shows that 0 ; :::; r 1 is also a basis. Now .!/  D
and ! 0 .D/ D 0 are both obviously equivalent to a0 D a1 D    D ar 1 D 0. The proof
for other divisors is similar.
    We finally prove the exactness of the sequence in (5.1). The injectivity of .d i/D follows
from the fact that iW F ,! C .r/ is a closed immersion. Moreover the sequence is a complex
because f ı i is the constant map x 7! a. It remains to show that

                             dim Im.d i/D D dim Ker.df .r/ /D :

    Identify Ta .J /_ with .C; ˝ 1 / using the isomorphisms arising from (2.1). Then (5.3)
shows that ! is zero on the image of TD .C .r/ / if and only if .!/  D, that is, ! 2
  .C; ˝ 1 . D//. Therefore the image of .df .r/ /D has dimension g h0 .˝ 1 . D// D
g h1 .D/, and so its kernel has dimension r g C h1 .D/. On the other hand, the image
of .d i /D has dimension jDj. The Riemann-Roch theorem says precisely that these two
numbers are equal, and so completes the proof.

C OROLLARY 5.5. For all r  g, f r W C r ! W r has degree rŠ.

P ROOF. It is the composite of W C r ! C .r/ and f .r/ .                                     2


R EMARK 5.6. (a) The theorem shows that J is the unique abelian variety birationally
equivalent to C .g/ . This observation is the basis of Weil’s construction of the Jacobian.
(See 7.)
    (b) The exact sequence in (5.1b) can be regarded as a geometric statement of the
Riemann-Roch theorem (see especially the end of the proof). In fact it is possible to prove
the Riemann-Roch theorem this way (see Mattuck and Mayer 1963).
    (c) As we observed above, the fibre of f .r/ W C .r/ .k/ ! J.k/ containing D can be
identified with the linear system jDj. More precisely, the fibre of the map of functors
C .r/ ! J is the functor DivD  C of (3.14); therefore the fibre of f
                                                                      .r/ containing D (in the

sense of algebraic spaces) is a copy of projective space of dimension h0 .D/ 1. Corollary
3.10 of Chapter I shows that conversely every copy of projective space in C .r/ is contained
in some fibre of f .r/ . Consequently, the closed points of the Jacobian can be identified with
the set of maximal subvarieties of C .r/ isomorphic to projective space.
    Note that for r > 2g 2, jDj has dimension r g, and so .df .r/ /D is surjective, for
all D. Therefore f .r/ is smooth (see Hartshorne 1977, III 10.4), and the fibres of f .r/ are
precisely the copies of Pr g contained in C .r/ . This last observation is the starting point of
Chow’s construction of the Jacobian Chow 1954.
104                                                 CHAPTER III. JACOBIAN VARIETIES


6     The Jacobian variety as Albanese variety; autoduality
Throughout this section C will be a complete nonsingular curve of genus g > 0 over a field
k, and J will be its Jacobian variety.

P ROPOSITION 6.1. Let P be a k-rational point on C . The map f P W C ! J has the
following universal property: for any map 'W C ! A from C into an abelian variety sending
P to 0, there is a unique homomorphism W J ! A such that ' D ı f P .

P ROOF. Consider the map
                                                X
                           .P1 ; :::; Pg / 7!       .Pi /W C g ! A:
                                                i

Clearly this is symmetric, and so it factors through C .g/ . It therefore defines a rational map
  W J ! A, which (I 3.2) shows to be a regular map. It is clear from the construction that
   ı f P D ' (note that f P is the composite of Q 7! Q C .g 1/P W C ! C .g/ with
f W C .g/ ! J /. In particular,
  .g/                                   maps 0 to 0, and (I 1.2) shows that it is therefore a
homomorphism. If     0  is a second homomorphism such that 0 ı f P D ', then and 0
agree on f .C / C    C f P .C / (g copies), which is the whole of J .
            P
                                                                                               2


C OROLLARY 6.2. Let N be a divisorial correspondence between .C; P / and J such that
.1  f P / N  LP ; then N  MP (notations as in 2 and (1.7)).

P ROOF. Because of (I 5.13), we can assume k to be algebraically closed. According to
(1.7) there is a unique map 'W J ! J such that N  .1  '/ MP . On points ' is the map
sending a 2 J.k/ to the unique b such that MP jC  fbg  N jC  fag. By assumption,

                  N jC  ff P Qg  LP jC  fQg  MP jC  ff P Qg,

and so .' ı f P /.Q/ D f P .Q/ for all Q. Now (6.1) shows that f is the identity map.         2


C OROLLARY 6.3. Let C1 and C2 be curves over k with k-rational points P1 and P2 , and let
J1 and J2 be their Jacobians. There is a one-to-one correspondence between Homk .J1 ; J2 /
and the set of isomorphism classes of divisorial correspondences between .C1 ; P1 / and
.C2 ; P2 /.

P ROOF. A divisorial correspondence between .C2 ; P2 / and .C1 ; P1 / gives rise to a mor-
phism .C1 ; P1 / ! J2 (by 1.7), and this morphism gives rise to homomorphism J1 ! J2
(by 6.1). Conversely, a homomorphism W J1 ! J2 defines a divisorial correspondence
.1  .f P1 ı // MP2 between .C2 ; P2 / and .C1 ; P1 /.                                 2


     In the case that C has a point P rational over k, define F W C  C ! J to be the map
.P1 ; P2 / 7! f P .P1 / f P .P2 /. One checks immediately that this is independent of the
choice of P . Thus, if P 2 C.k 0 / for some Galois extension k 0 of k, and F W Ck 0 Ck 0 ! Jk 0
is the corresponding map, then F D F for all  2 Gal.k 0 =k/ ; therefore F is defined
over k whether or not C has a k-rational point. Note that it is zero on the diagonal  of
C  C.
6. THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY                                105


P ROPOSITION 6.4. Let A be an abelian variety over k. For any map 'W C  C ! A such
that './ D 0, there is a unique homomorphism W J ! A such that ı F D '.

P ROOF. Let k 0 be a finite Galois extension of k, and suppose that there exists a unique
homomorphism W Ck 0 ! Jk 0 such that ı Fk 0 D 'k 0 . Then the uniqueness implies
that  D       for all  in Gal.k 0 =k/, and so    is defined over k. It suffices therefore
to prove the proposition after extending k, and so we can assume that C has a k-rational
point P . Now (I 1.5) shows that there exist unique maps '1 and '2 from C to A such that
'1 .P / D 0 D '2 .P / and '.a; b/ D '1 .a/C'2 .b/ for all .a; b/ 2 C C . Because ' is zero
on the diagonal, '1 D '2 . From (6.1) we know that there exists a unique homomorphism
   from J to A such that '1 D ı f , and clearly is also the unique homomorphism such
that ' D ı F .                                                                            2


R EMARK 6.5. The proposition says that .A; F / is the Albanese variety of C in the sense
of Lang 1959, II 3, p45. Clearly the pairs .J; f P / and .J; F / are characterized by the
universal properties in (6.1) and (6.4).

     Assume again that C has a k-rational point P , and let  D W g 1 . It is a divisor on
J , and if P is replaced by a second k-rational point,  is replaced by a translate. For any
effective divisor D on J , write

    L0 .D/ D m L.D/ ˝ p  L.D/ 1 ˝ q  L.D/ 1 D L.m 1 .D/            D  J _ J  D/.

Recall (I 8.1 et seqq.), that D is ample if and only if 'L.D/ W J ! J _ is an isogeny, and
then .1  'L.D/ / .P/ D L0 .D/, where P is the Poincaré sheaf on J  J _ . Write 
for the image of  under the map . 1/J W J ! J , and a for ta  D  C a, a 2 J.k/.
Abbreviate . /a by a .

T HEOREM 6.6. The map 'L./ W J ! J _ is an isomorphism; therefore, 1  'L./ is an
isomorphism .J  J; L0 .// ! .J  J _ ; P/.

P ROOF. As usual, we can assume k to be algebraically closed. Recall (Milne 1986, 12.13)
that 'L. / D . 1/2 'L./ D 'L./ , and that 'L.a / D 'L./ for all a 2 J.k/.        2


L EMMA 6.7. Let U be the largest open subset of J such that
  (i) the fibre of f .g/ W C .g/ ! J at any point of U has dimension zero, and
 (ii) if a 2 U.k/ and D.a/ is the unique element of C .r/ .k/ mapping to a, then D.a/ is a
      sum of g distinct points of C.k/.
Then f   1 . / D D.a/ (as a Cartier divisor) for all a 2 U.k/ , where f   D f PWC ! J.
             a


P ROOF. Note first that U can be obtained by removing the subset over which the fibres
have dimension > 0, which is closed (AG 10.9), together with the images of certain closed
subsets of the form   C g 2 . These last sets are also closed because C g ! J is proper
(AG Chapter 7), and it follows that U isPa dense open subset of J .
    Let a 2 U.k/, and let D.a/ D          i Pi , Pi ¤ Pj for i P  ¤ j . A point Q1 of C
                                                                    g
maps to a point    of  a if and only if there   exists a divisor   i D2 Qi on C such that
  P
               P      P
                                                          P  g
f .Q/ D           i f .Qi / C a. The equality implies        iD1 Qi  D, and the fact that
106                                                CHAPTER III. JACOBIAN VARIETIES

                                                                                       1 . / is
                                      P
jDj has dimension 0 implies that        i Qi D D. It follows that the support of f         a
fP1 ; :::; Pg g, and it remains to show that f 1 .a / has degree  g for all a.
    Consider the map W C   ! J sending .Q; b/ to f .Q/ C b. As the composite of
   with 1  f g 1 W C  C g 1 ! C   is f g W C g ! J , and these maps have degrees
.g 1/Š and gŠ respectively (5.5), has degree g. Also is projective because C   is a
projective variety (see Hartshorne 1977, II, Ex. 4.9). Consider a 2 U ; the fibre of over a
is f 1 .a / (more accurately, it is the algebraic subspace of C associated with the Cartier
divisor a /. Therefore the restriction of to         1 .U / is quasi-finite and projective, and

so is finite (AG 8.19). As U is normal, this means that all the fibres of over points of U
are finite schemes of rank  g (AG 10.12). This completes the proof of the lemma.              2


L EMMA 6.8. (a) Let a 2 J.k/, and let f .g/ .D/ D a; then f  L.a /  L.D/.
   (b) The sheaves .f  . 1/J / L0 . / and MP on C  J are isomorphic.

P ROOF. Note that (6.7) shows that the isomorphism in (a) holds for all a in a dense open
subset of J . Note also that the map
                          Q7!.Q;a/             f . 1/             m
                    C           ! C  fag           ! J J           ! J
equals t a ı f , and so

   .f  . 1// m L. /jC  fag ' L.t a1  /jf .C / ' L.a /jf .C /  f  L.a /:

Similarly
                        .f  . 1// p  L. /jC  fag ' f  L. /,
and
                               .f  . 1// q  L. /jC  fag
is trivial. On the other hand, MP is an invertible sheaf on C  J such that
     (i) MP jC  fag  L.D gP / if D is an effective divisor of degree g on C such that
f .g/ .D/ D aI
     (ii) MP jfP g  J is trivial.
     Therefore (a) is equivalent to .f . 1// m L. /jC fag being isomorphic to MP ˝
  
p L.gP /jC  fag for all a. As we know this is true for all a in a dense subset of J , (I 5.19)
applied to
                        MP ˝ p  L.gP / ˝ .f  . 1// m L. / 1
proves (a). In particular, on taking a D 0, we find that f  L. /  L.gP /, and so
.f  . 1// p  L. /  p  L.gP /. Now (I 5.16) shows that

                 .f  . 1// .m L. / ˝ p  L. / 1 /  MP ˝ q  N

for some invertible sheaf N of J . On computing the restrictions of the sheaves to fP g  J ,
we find that N  . 1/ L. /, which completes the proof.                                   2


   Consider the invertible sheaf .f  1/ P on C  J _ . Clearly it is a divisorial correspon-
dence, and so there is a unique homomorphism f _ W J _ ! J such that .1  f _ / MP 
.f  1/ P. The next lemma completes the proof of the theorem.

L EMMA 6.9. The maps f _ W J _ ! J and 'L./ W J ! J _ are inverse.
6. THE JACOBIAN VARIETY AS ALBANESE VARIETY; AUTODUALITY                                   107


P ROOF. Write     D      'L./ D    'L.    / . We have

                  .1     / .1  f _ / MP  .1         / .f  1/ P
                                               .f       / P
                                               .f  . 1// .1  'L./ / P
                                               .f  . 1// L0 . /
                                               MP :

Therefore, f _ ı is a map ˛W J ! J such that .1  ˛/ MP  MP ; but the only map
with this property is the identity.                                            2


R EMARK 6.10. (a) Lemma 6.7 shows that f .C / and  cross transversely at any point
of U. This can be proved more directly by using the descriptions of the tangent spaces
implicitly given near the end of the proof of (5.1).
   (b) In (6.8) we showed that MP  .f  . 1// L0 . /. This implies

                            MP  .f  . 1// .1  'L. / / P
                                    .f  . 1// .1  'L./ / P
                                    .f  . 1// L0 ./:

Also, because D 7! 'L.D/ is a homomorphism, 'L. / D                'L./ , and so

                            MP  .f  . 1// .1  'L./ / P
                                    .f  1/ .1  'L. / / P
                                    .f  1/ L0 . /:

    (c) The map on points J _ .k/ ! J.k/ defined by f _ is induced by f  W Pic.J / !
Pic.C /.
    (d) Lemma 6.7 can be generalized as follows. An effective canonical divisor K defines
a point on C .2g 2/ whose image in J will P be denoted . Let a be a point of J such thatr
a  is not in .W g 2 / , and write a D i f .Pi / with P P1 ; :::; Pg points on C . Then W
and .W  g r                             r
            /a intersect properly, and W  .W g  r /a D .wi1 :::ir / where

                              wi1 :::ir D f .Pi1 / C    C f .Pir /

and the sum runs over the .gr / combinations obtained by taking r elements from f1; 2; :::; gg.
See Weil 1948b, 39, Proposition 17.

S UMMARY 6.11. Between .C; P / and itself, there is a divisorial correspondence LP D
L. fP g  C C  fP g/:
     Between .C; P / and J there is the divisorial correspondence MP ; for any divisorial
correspondence L between .C; P / and a pointed k-scheme .T; t/, there is a unique mor-
phism of pointed k-schemes 'W T ! J such that .1  '/ MP  LP . In particular, there
is a unique map f P W C ! J such that .1  f P / MP  LP and f .P / D 0.
     Between J and J _ there is a canonical divisorial correspondence P (the Poincaré
sheaf); for any divisorial correspondence L between J and a pointed k-scheme .T; t/ there
is a unique morphism of pointed k-schemes W T ! J such that .1  / P  L.
108                                                 CHAPTER III. JACOBIAN VARIETIES


     Between J and J there is the divisorial correspondence L0 ./. The unique morphism
J ! J _ such that .1  / P  L0 ./ is 'L./ , which is an isomorphism. Thus 'L./
is a principal polarization of J , called the canonical polarization. There are the following
formulas:
                      MP  .f  . 1// L0 ./  .f  1/ L0 ./ 1 :
Consequently,
                                  LP  .f  f / L0 ./ 1 :
If f _ W J _ ! J is the morphism such that .f  1/ P  .1  f _ / MP , then f _ D
     1
  'L./  .

E XERCISE 6.12. It follows from (6.6) and the Riemann-Roch theorem (I 11.1) that . g / D
gŠ. Prove this directly by studying the inverse image of  (and its translates) by the map
C g ! J . (Cf. AG 12.10 but note that the map is not finite.) Hence deduce another proof
of (6.6).


7      Weil’s construction of the Jacobian variety
As we saw in (5.6a), the Jacobian J of a curve C is the unique abelian variety that is
birationally equivalent to C .g/ . To construct J , Weil used the Riemann-Roch theorem to
define a rational law of composition on C .g/ and then proved a general theorem that allowed
him to construct an algebraic group out of C .g/ and the rational law. Finally, he verified
that the algebraic group so obtained had the requisite properties to be called the Jacobian of
C . We give a sketch of this approach.
    A birational group over k (or a nonsingular variety with a normal law of composition
in the terminology of Weil 1948b, V) is a nonsingular variety V together with a rational
map mW V  V _ _ _/ V such that
    (a) m is associative (that is, .ab/c D a.bc/ whenever both terms are defined);
    (b) the rational maps .a; b/ 7! .a; ab/ and .a; b/ 7! .b; ab/ from V  V to V  V are
        both birational.
      Assume that C has a k-rational point P .
L EMMA 7.1. (a) There exists an open subvariety U of C .g/  C .g/ such that, for all fields
K containing k and all .D; D 0 / in U.K/, h0 .D C D 0 gP / D 1.
    (b) There exists an open subset V of C .g/  C .g/ such that for all fields K containing
k and all .D; D 0 / in V .K/, h0 .D 0 D C gP / D 1.

P ROOF. (a) Let Dcan be the canonical relative effective divisor on C  C .2g/ =C .2g/ con-
structed in 3. According to the Riemann-Roch theorem, h0 .D gP /  1 for all divisors
of degree 2g on C , and so (I 4.2) shows that the subset U of C .2g/ of points t such that
h0 ..Dcan /t gP / D 1 is open. On the other hand, (5.2b) shows that there exist positive
divisors D of degree g such that h0 ..D C gP / gP / D 1, and so U is nonempty. Its
inverse image in C .g/  C .g/ is the required set.
    (b) The proof is similar to that of (a): the Riemann-Roch theorem shows that h0 .D 0
D C gP /  1 for all D and D 0 , we know there exists a D 0 such that h0 .D 0 gP C gP / D
h0 .D 0 / D 1, and (I 4.2) applied to the appropriate invertible sheaf on C  C .r/  C .r/ gives
the result.                                                                                     2
7. WEIL’S CONSTRUCTION OF THE JACOBIAN VARIETY                                               109


P ROPOSITION 7.2. There exists a unique rational map

                                  mW C .g/  C .g/ _ _ _/ C .g/

whose domain of definition contains the subset U of (7.1a) and which is such that for all
fields K containing k and all .D; D 0 / in U.K/, m.D; D 0 /  D C D 0 gP ; moreover m
makes C .g/ into a birational group.

P ROOF. Let T be an integral algebraic space over k. If we identify C .g/ with the functor it
represents (see 3.13), then an element of U.T / is a pair of relative effective divisors .D; D 0 /
on C T =T such that, for all t 2 T , h0 .Dt CDt0 gP / D 1. Let L D L.DCD 0 gP T /.
Then (I 4.2d) shows that q .L/ is an invertible sheaf on T . The canonical map q  q L ! L
when tensored with .q  q L/ 1 gives a canonical global section sW OT ! L ˝ .q  q L/ 1 ,
which determines a relative effective divisor m.D; D 0 / of degree g on C  T =T (see 3.6).
The construction is clearly functorial. Therefore we have constructed a map mW U ! C .g/
as functors of integral schemes over k, and this is represented by a map of varieties. On
making the map explicit in the case that K is the spectrum of a field, one sees easily that
m.D; D 0 /  D C D 0 gP in this case.
    The uniqueness of the map is obvious. Also associativity is obvious since it holds on an
open subset of U.K/:
                             m..D; D 0 /; D 00 / D m.D; .D 0 ; D 00 //
because each is an effective divisor on C linearly equivalent to D C D 0 C D 00 2gP , and
in general h0 .D C D 0 C D 00 2gP / D 1.
     A similar argument using (7.1b) shows that there is a map rW V ! C .g/ such that .p; r/
is a birational inverse to

                     .a; b/ 7! .a; ab/W C .g/  C .g/ _ _ _/ C .g/  C .g/ .

Because the law of composition is commutative, this shows that .a; b/ 7! .b; ab/ is also
birational. The proof is complete.                                                     2


T HEOREM 7.3. For any birational group V over k, there is a group variety G over k and a
birational map f W V _ _ _/ G such that f .ab/ D f .a/f .b/ whenever ab is defined; more-
over, G is unique up to a unique isomorphism.

P ROOF. In the case that V .k/ is dense in V (for example, k is separably closed), this is
proved in Artin 1986,2. (Briefly, one replaces V by an open subset where m has better
properties, and obtains G by patching together copies of translates of U by elements of
V .k/ .) From this it follows that, in the general case, the theorem holds over a finite Galois
extension k 0 of k. Let  2 Gal.k 0 =k/. Then f W Vk 0 _ _ _/ G is a birational map, and as
Vk 0 D Vk 0 , the uniqueness of G shows that there is a unique isomorphism ' W G ! G
such that ' ı f D f . For any ;  2 Gal.k 0 =k/;

                .' ı  ' / ı . f / D ' ı .' ı f / D f D '  ı f;

and so ' ı  ' D ' . Descent theory (see 1.13) now shows that G is defined over k.          2


    Let J be the algebraic group associated by (7.3) to the rational group defined in (7.2).
110                                               CHAPTER III. JACOBIAN VARIETIES


P ROPOSITION 7.4. The variety J is complete.

P ROOF. This can be proved using the valuative criterion of properness. (For Weil’s original
account, see Weil 1948b, Théoreme 16, et seqq..)                                         2


C OROLLARY 7.5. The rational map f W C .g/ _ _ _/ J is a morphism. If D and D 0 are
linearly equivalent divisors on CK for some field K containing k, then f .D/ D f .D 0 /.

P ROOF. The first statement follows from (I 3.2). For the second, recall that if D and D 0 are
linearly equivalent then they lie in a copy of projective space contained in C .g/ (see 3.14).
Consequently (I 3.10) shows that they map to the same point in J .                           2


      We now prove that J has the correct universal property.

T HEOREM 7.6. There is a canonical isomorphism of functors W PC0 ! J .

P ROOF. As in 4, it suffices to show that PCr is representable by J for some r. In this
case we take r D g. Let L be an invertible sheaf with fibres of degree g on C  T . If
dimk .Ct ; Lt / D 1 for some t, then this holds for all points in an open neighbourhood Ut
of t. As in the proof of (7.2), we get a relative effective divisor sW OS ! L ˝ .q  q L/ 1
of degree g on Ut . This family of Cartier divisors defines a map Ut ! C .g/ which, when
composed with f , gives a map L W Ut ! J . On the other hand, if dimk .Ct ; Lt / > 1,
then we choose an invertible sheaf L0 of degree zero on C such that dim .Ct ; Lt ˝ L0 / D
1, and define L W Ut ! C .g/ on a neighbourhood of t to be the composite of L˝p L0 with
t a , where a D f .D/ for D an effective divisor of degree g such that L.D gP /  L0 .
One checks that this map depends only on L, and that the maps for different t agree on the
overlaps of the neighbourhoods. They therefore define a map T ! J .                        2


R EMARK 7.7. Weil of course did not show that the Jacobian variety represented a functor
on k-schemes. Rather, in the days before schemes, the Jacobian variety was characterized
                                                                                           '
by the universal property in (6.1) or (6.4), and shown to have the property that Pic0 .C / !
J.k/. See Weil 1948b or Lang 1959.


8      Generalizations
It is possible to construct Jacobians for families of curves. Let W C ! S be projective flat
morphism whose fibres are integral curves. For any S-scheme T of finite-type, define

                    PCr .T / D fL 2 Pic.C S T / j deg.Lt / D r all tg= 

where L  L0 if and only if L  L0 ˝ q  M for some invertible sheaf M on T. (The
degree of an invertible sheaf on a singular curve is defined as in the nonsingular case: it is
the leading coefficient of .C; Ln / as a polynomial in n/. Note that PCr is a functor on the
category of S -schemes of finite-type.

T HEOREM 8.1. Let W C ! S be as above; then there is a group scheme J over S with
connected fibres and a morphism of functors PC0 ! J such that PC0 .T / ! J .T / is always
injective and is an isomorphism whenever C S T ! T has a section.
8. GENERALIZATIONS                                                                                    111


    In the case that S is the spectrum of a field (but C may be singular), the existence
of J can be proved by Weil’s method (see Serre 1959, V). When C is smooth over S ,
one can show as in 3 that C .r/ (quotient of C S ::: S C by Sr / represents the functor
DivrC=S sending an S-scheme T to the set of relative effective Cartier divisors of degree r
on C S T =T . In general one can only show more abstractly that DivrC=S is represented
by a Hilbert scheme. There is a canonical map DivrC=S ! PC=S  r
                                                                 and the second part of the
                                          r                   r
proof deduces the representability of PC=S from that of DivC=S . (The only reference for
the proof in the general case seems to be Grothendieck’s original rather succinct account
Grothendieck 62, 232;2 we sketch some of its ideas below.)
    As in the case that the base scheme is the spectrum of a field, the conditions of the
theorem determine J uniquely; it is called the Jacobian scheme of C=S. Clearly J com-
mutes with base change: the Jacobian of C S T over T is J S T . In particular, if Ct is a
smooth curve over k.t /, then Jt is the Jacobian of Ct in the sense of 1. Therefore if C is
smooth over S, then J is an abelian scheme, and we may think of it as a family of Jacobian
varieties. If C is not smooth over S , then J need not be proper, even in the case that S is
the spectrum of a field.

E XAMPLE 8.2. Let C be complete smooth curve over an algebraically
                                                            P           closed field k. By a
modulus for C one simply means an effective divisor m D P nP P on C . Let m be such
a modulus, and assume that deg.m/  2. We shall associate with C and m a new curve Cm
having a single singularity at a point to be denoted by Q. The underlying topological space
of Cm is .C S / [ fQg, where S is the support of m. Let OQ D k C cQ , where

                       cQ D ff 2 k.C / j ord.f /  nP all P in S g;
                                                         T
and define OCm to be the sheaf such that .U; OCm / D P 2U OP . The Jacobian scheme
Jm of Cm is an algebraic group over k called the generalized Jacobian of C relative to
m. By definition, Jm .K/ is the group of isomorphism classes of invertible sheaves on Cm
of degree 0. It can also be described as the group of divisors of degree 0 on C relatively
prime to m, modulo the principal divisors defined by elements congruent to 1 modulo m
(an element of k.C / is congruent to 1 modulo if ordP .f 1/  nP for all P in S ). For
each modulus m with support on S there is a canonical map fm W C r S ! Jm , and these
maps are universal in the following sense: for any morphism f W C r S ! G from C r S
into an algebraic group, there is a modulus m and a homomorphism 'W Jm ! G such that
f is the composite of fm ı ' with a translation. (For a detailed account of this theory, see
Serre 1959.)

     We now give a brief sketch of part of Grothendieck’s proof of (8.1). First we need the
notion of the Grassmann scheme.
     Let E be a locally free sheaf of OS -modules of finite rank, and, for an S-scheme T of
finite-type, define GrassEn .T / to be the set of isomorphism classes of pairs .V; h/, where
V is a locally free OT -module of rank n and h is an epimorphism OT ˝k E  V. For
example, if E D OSm , then GrassEn .T / can be identified with the set of isomorphism classes
of pairs .V; .e1 ; :::; em // where V is a locally free sheaf of rank n on T and the ei are
sections of V over T that generate V; two such pairs .V; .e1 ; :::; em // and .V 0 ; .e10 ; :::; em
                                                                                                  0 //


    2 See also:Fantechi, Barbara; Göttsche, Lothar; Illusie, Luc; Kleiman, Steven L.; Nitsure, Nitin; Vis-
toli, Angelo. Fundamental algebraic geometry. Grothendieck’s FGA explained. Mathematical Surveys and
Monographs, 123. American Mathematical Society, Providence, RI, 2005. x+339 pp.
112                                                 CHAPTER III. JACOBIAN VARIETIES

                                                
are isomorphic if there is an isomorphism V ! V 0 carrying each ei to ei0 . In particular,
        N C1
GrassO1      .T / D PN
                     S .T / (cf. Hartshorne 1977, II 7.1).
P ROPOSITION 8.3. The functor T 7! GrassnE .T / is representable by a projective variety
GnE over S.

P ROOF. The construction of GnE is scarcely more difficult than that of PN
                                                                         S (see Grothendieck
and Dieudonné 1971, 9.7).                                                                2


     Choose an r > 2g 2 and an m > 2g 2 C r. As in the case that S is the spectrum of
a field, we first need to construct the Jacobian under the assumption that there is a section
sW S ! C. Let E be the relative effective divisor on C=S defined by s (see 3.9), and for any
invertible sheaf L on C S T , write L.m/ for L ˝ L.mE/. The first step is to define an
embedding of DivrC=S into a suitable Grassmann scheme.
     Let D 2 DivrC=S .T /, and consider the exact sequence
                           0 ! L. D/ ! OCT ! OD ! 0
on C S T (we often drop the S from C S T ). This gives rise to an exact sequence
                     0 ! L. D/.m/ ! OCT .m/ ! OD .m/ ! 0;
and on applying q we get an exact sequence
      0 ! q L. D/.m/ ! q OC T .m/ ! q OD .m/ ! R1 q L. D/.m/ ! ::::
Note that, for all t in T , H 1 .Ct ; L. D/.m// is dual to H 0 .Ct ; L.K C D mEt //, where
Et is the divisor s.t / of degree one on Ct . Because of our assumptions, this last group is
zero, and so (I 4.2e) R1 q L. D/.m/ is zero and we have an exact sequence
                 0 ! q L. D/.m/ ! q OCT .m/ ! q OD .m/ ! 0:
Moreover q OD .m/ is locally free of rank r, and q .OCT .m// D q OC .m/ ˝ OT (loc.
                                                                q O .m/
cit.), and so we have constructed an element ˚.D/ of Grassr  C .T /.
                                                                                q O .m/
     On the other hand, suppose a D .q OCT .m/  V/ is an element of Grassn C .T /.
If K is the kernel of q  q OCT .m/  q  V, then K. m/ is a subsheaf of q  q OCT , and
its image under q  q OCT ! OCT is an ideal in OCT . Let .a/ be the subscheme
associated to this ideal. It is clear from the constructions that ˚.D/ D D for any relative
divisor of degree r. We have a diagram of natural transformations
                     ˚
          DivrC .T / ! Grassqn OC .m/ .T / ! S.T /  DivrC .T /;        ˚ D id;
where S.T / denotes the set of all closed subschemes of C S T . In particular, we see that
˚ is injective.
                                                                                     q O .m/
P ROPOSITION 8.4. The functor ˚ identifies DivrC with a closed subscheme of Grassn C           .

P ROOF. See Grothendieck 62, p221-12, (or, under different hypotheses, Mumford 1970,
Lecture 15).                                                                      2


    Finally one shows that the fibres of the map DivrC=S ! PC=S  r
                                                                     are represented by the
projective space bundles associated with certain sheaves of OS -modules (Grothendieck 62,
                                                           r
p232-11; cf. (5.6c)) and deduces the representability of PC=S (loc. cit.).
9. OBTAINING COVERINGS OF A CURVE FROM ITS JACOBIAN                                                           113


9     Obtaining coverings of a curve from its Jacobian; application
      to Mordell’s conjecture
Let V be a variety over field k, and let W W ! V be a finite étale map. If there is a finite
group G acting freely on W by V -morphisms in such a way that V D GnW , then .W; /
is said to be Galois covering3 of V with Galois group G. When G is abelian, .W; / is
said to be an abelian covering of V . Fix a point P on V . Then the Galois coverings of
V are classified by the (étale) fundamental group 1 .V; P / and the abelian coverings by
the maximal abelian quotient 1 .V; P /ab of 1 .V; P /. For any finite abelian group M ,
Hom.1 .V; P /; M / (set of continuous homomorphisms) is equal to the set of isomorphism
classes of Galois coverings of V with Galois group M . If, for example, V is nonsingular
and we take P to be the generic point of V , then every finite connected étale covering of V
is isomorphic to the normalization of V in some finite extension K 0 of k.P / contained in
a fixed separable algebraic closure K sep of K; moreover, 1 .V; P / D Gal.K un =K/ where
K un is the union of all finite extensions K 0 of k.P / in K sep such that the normalization
of V in K 0 is étale over V. The covering corresponding to a continuous homomorphism
˛W Gal.K un =K/ ! M is the normalization of V in .K sep /Ker.˛/ . (See LEC, 3, or Milne
1980, I 5, for a more detailed discussion of étale fundamental groups.)
     Now let C be a complete nonsingular curve over a field k, and let f D f P for some
P in C.k/. From a finite étale covering J 0 ! J of J , we obtain an étale covering of C by
pulling back relative to f :

                                    J0             C0              C J J 0
                                    ?              ?
                                    ?              ?
                                    y              y
                                            f
                                     J             C:
Because all finite étale coverings of J are abelian (cf. I 12.3), we only obtain abelian
coverings of C in this way. The next proposition shows that we obtain all such coverings.
   Henceforth, k will be separably closed.

P ROPOSITION 9.1. If J 0 ! J is a connected étale covering of J , then C 0 D C J J 0 ! C
is a connected étale covering of C , and every connected abelian covering of C is obtained in
this way. Equivalently, the map 1 .C; P /ab ! 1 .J; 0/ induced by f P is an isomorphism.

P ROOF. The equivalence of the two assertions follows from the interpretation of

                                            Hom.1 .V; P /; M /

recalled above and the fact that 1 .J; 0/ is abelian. We shall prove the second assertion.
For this it suffices to show that for all integers n, the map

                         Hom.1 .J; 0/; Z=nZ/ ! Hom.1 .C; P /; Z=nZ/

induced by f P is an isomorphism. The next two lemmas take care of the case that n is
prime to the characteristic of k.                                                  2

    3 Some authors call a finite covering W ! V is Galois if the field extension k.W /=k.V / is Galois, i.e., if

it is generically Galois, but this conflicts with Grothendieck’s terminology and is not the natural definition.
114                                               CHAPTER III. JACOBIAN VARIETIES


L EMMA 9.2. Let V be complete nonsingular variety and let P be a point of V ; then for all
integers n prime to the characteristic of k,

                         Homconts .1 .V; P /; Z=nZ/ ' Pic.V /n :

P ROOF. Let D be a (Weil) divisor on V such that nD D .g/ for some g 2 k.V /, and
let V 0 be the normalization of V in the Kummer extension k.V /.g 1=n / of k.V /. A purity
theorem Grothendieck 1971, X 3.1, shows that V 0 ! V is étale if, for all prime divisors
Z on V , the discrete valuation ring OZ (local ring at the generic point of Z) is unramified
in k.V 0 /. But the extension k.V 0 /=k.V / was constructed by extracting the nth root of an
element g such that ordZ .g/ D 0 if Z is not in the support of D and is divisible by n
otherwise, and it follows from this that OZ is unramified. Conversely, let V 0 ! V be a
Galois covering with Galois group Z=nZ. Kummer theory shows that the k.V 0 /=k.V / is
obtained by extracting the nth root of an element g of k.V /. Let Z be a prime divisor on
V. Because OZ is unramified in k.V 0 /, ordZ .g/ must be divisible by n (or is zero), and so
.g/ D nD for some divisor D. Obviously D represents an element of Pic.V /n . It is easy
to see now that the correspondence we have defined between coverings of V and elements
of Pic.V /n is one-to-one. (For a proof using étale cohomology, see Milne 1980, III, 4.11.)2

L EMMA 9.3. The map Pic.J / ! Pic.C / defined by f induces an isomorphism Pic0 .J / !
Pic0 .C /.

P ROOF. This was noted in (6.10c).                                                         2


    In the case that n D p D char.k/, (9.2) and (9.3) must be replaced by the following
analogues.

L EMMA 9.4. For any complete nonsingular variety V and point P ,

          Hom.1 .V; P /; Z=pZ/ ' Ker.1         F W H 1 .V; OV / ! H 1 .V; OV //;

where F is the map induced by a 7! ap W OV ! OV .

P ROOF. See Milne 1980, p127, for a proof using étale cohomology as well as for hints for
an elementary proof.                                                                    2


L EMMA 9.5. The map f P W C ! J induces an isomorphism H 1 .J; OJ / ! H 1 .C; OC /.

P ROOF. See Serre 1959, VII, Théoreme 9. (Alternatively, note that the same argument as
                                                            '
in the proof of (2.1) gives an isomorphism H 1 .J; OJ /     ! T0 .J _ /, and we know that
J ' J _ .)                                                                                 2


    To prove the case n D p m , one only has to replace OC and OJ by the sheaves of Witt
vectors of length m, Wm OC and Wm OJ . (It is also possible to use a five-lemma argument
starting from the case m D 1:)

C OROLLARY 9.6. For all primes `, the map of étale cohomology groups H 1 .J; Z` / !
H 1 .C; Z` / induced by f is an isomorphism.
9. OBTAINING COVERINGS OF A CURVE FROM ITS JACOBIAN                                        115


P ROOF. For any variety V , H 1 .Vet ; Z=nZ/ D Hom.1 .V; P /; Z=nZ/ Milne 1980, III.4.
Therefore, there are isomorphisms
                                         '
                       H 1 .J; Z=`m Z/ ! Hom.1 .J; P /; Z=`m Z/
                                         '
                                         ! Hom.1 .C; P /; Z=`m Z/
                                         '
                                         ! H 1 .C; Z=`m Z/;

and we obtained the required isomorphism by passing to the limit.                            2


    To obtain ramified coverings of C , one can use the generalized Jacobians.
P ROPOSITION 9.7. Let C 0 ! C be a finite abelian covering of C that is unramified outside
a finite set ˙. Then there is a modulus m with support on ˙ and an étale isogeny J 0 ! Jm
whose pull-back by fm is C 0 r f 1 .˙/.

P ROOF. See Serre 1959.                                                                      2


E XAMPLE 9.8. In the case that the curve is P1 and m D 0 C 1 , we have Jm D
P1 r f0; 1g, which is just the multiplicative group GL1 , and fm is an isomorphism. For
any n prime to the characteristic, there is a unique unramified covering of P1 r f0; 1g of
degree n, namely multiplication by n on P1 r f0; 1g. When k D C, this covering is the
usual unramified covering z 7! z n W C r f0g ! C r f0g.

P ROPOSITION 9.9. Let C be a curve of genus g over a number field k, and let P be a k-
rational point of C . Let S be a finite set of primes of k containing all primes dividing 2 and
such that C has good reduction outside S . Then there exists a field k 0 of degree  22g over
                                                                                2g
k and unramified over S, and a finite map fP W CP ! Ck 0 of degree  22 .g 1/C2gC1 ,
ramified exactly over P , and such that CP has good reduction outside S.

P ROOF (S KETCH ) Let C 0 be the pull-back of 2W J ! J ; it is an abelian étale covering of
C of degree 22g , and the Hurwitz genus formula (Hartshorne 1977, IV 2.4) shows that the
genus g 0 of C 0 satisfies
                                2g 0 2 D 22g .2g 2/;
so that g 0 D 22g .g 1/ C 1. Let D be the inverse image of P on C 0 . It is a divisor of
degree 22g on C 0 , and after an extension k 0 of k of degree  22g unramified over S , some
point P of D will be rational. Let m D D P , and let C 00 be the pull-back of the covering
                                 0
2W Jm ! Jm (of degree  22g ) by C r ˙ ! Jm , where ˙ D Supp.D/ r fP g. Then C 00
is a curve over k 0 , and we take CP to the associated complete nonsingular curve.         2


    This result has a very striking consequence. Recall that a conjecture of Shafarevich
states the following:
 9.10. For any number field k, integer g, and finite set S of primes of k, there are only
finitely many isomorphism classes of curves C of genus g over k having good reduction at
all primes outside S.

T HEOREM 9.11. Shafarevich’s conjecture (9.10) implies Mordell’s conjecture.
116                                                    CHAPTER III. JACOBIAN VARIETIES


P ROOF. Let C be curve of genus g  2 over k with good reduction outside a set S con-
taining all primes of k lying over 2. There is a finite field extension K of k containing all
extensions k 0 of k of degree  22g that are unramified outside S . For each k -rational point
P on C , Proposition (9.10) provides a map fP W CP ! CK of degree  a fixed bound
B.g/ which is ramified exactly over P ; moreover, CP has good reduction outside S . The
Hurwitz genus formula shows that
                           2g.CP /     2  B.g/.2g        2/ C B.g/      1:
Therefore Shafarevich’s conjecture implies that there can be only finitely many curves CP .
A classical result of de Franchis (Lang 1983, p223) states that for each CP , there are only
finitely many maps CP ! C (this is where it is used that g  2/. Therefore there can be
only finitely many of k-rational points on C , as predicted by Mordell.                    2



10     Abelian varieties are quotients of Jacobian varieties
The main result in this section sometimes allows questions concerning abelian varieties to
be reduced to the special case of Jacobian varieties.
T HEOREM 10.1. For any abelian variety A over an infinite field k, there is a Jacobian
variety J and a surjective homomorpism J  A.4

L EMMA 10.2. Let W W ! V be a finite morphism of complete varieties, and let L be an
invertible sheaf on V. If L is ample, then so also is   L.

P ROOF. We shall use the following criterion (Hartshorne 1977, III, 5.3): an invertible sheaf
L on a complete variety is ample if and only if, for all coherent OV -modules F, H i .V; F ˝
Ln / D 0 for all i > 0 and sufficiently large n. Also we shall need an elementary projection
formula: if N and M are coherent sheaves of modules on W and V respectively, then
                                  .N ˝   M/ '  N ˝ M:
(Locally, this says that if B is an A-algebra and N and M are modules over B and A
respectively, then N ˝B .B ˝A M / ' N ˝A M as A-modules.)
    Let F be a coherent OW -module. Because  is finite (hence affine), we have by
(Hartshorne 1977, II Ex.4.1, or Ex.8.2) that
                        H i .W; F ˝   Ln / ' H i .V;  .F ˝   Ln //.
The projection formula shows that the second group equals H i .V;  F ˝Ln /, which is zero
for all i > 0 and sufficiently large n because L is ample and  F is coherent (Hartshorne
1977, II 4.1). The criterion now shows that   L is ample.                               2


L EMMA 10.3. Let V be a nonsingular projective variety of dimension  2 over a field
k, and let Z be a hyperplane section of V relative to some fixed embedding V ,! Pn .
Then, for any finite map  from a nonsingular variety W to V ,  1 .Z/ is geometrically
connected (i.e., is connected and remains connected under extensions of the base field).
   4 This is true also over finite fields. See:

  Gabber, O. On space filling curves and Albanese varieties. Geom. Funct. Anal. 11 (2001), no. 6, 1192–
1200.
  Poonen, Bjorn, Bertini theorems over finite fields. Ann. of Math. (2) 160 (2004), no. 3, 1099–1127.
11. THE ZETA FUNCTION OF A CURVE                                                            117


P ROOF. The hypotheses are stable under a change of the base field, and so we can assume
that k is algebraically closed. It then suffices to show that  1 .Z/ is connected. Because
Z is an ample divisor on V , the preceding lemma shows that  1 .Z/ is the support of an
ample divisor on W , which implies that it is connected (Hartshorne 1977, III, 7.9).      2

P ROOF ( PROOF OF 10.1) Since all elliptic curves are their own Jacobians, we can assume
that dim.A/ > 1. Fix an embedding A ,! Pn of A into projective space. Then Bertini’s
theorem (Hartshorne 1977, II, 8.18) shows that there exists an open dense subset U of
the dual projective space Pn_     of PnN such that, for all hyperplanes H in U , AkN \ H is
                              kN      k
nonsingular and connected. Because k is infinite, U.k/ is nonempty (consider a line L in
Pn_  /, and so there exists such an H with coordinates in k. Then A \ H is a (geometrically
  kN
connected) nonsingular variety in Pn . On repeating the argument dim.A/ 1 times, we
arrive at a nonsingular curve C on A that is the intersection of A with a linear subspace of
Pn . Now (10.3) applied several times shows that for any nonsingular variety W and finite
map W W ! A,  1 .C / is geometrically connected.
     Consider the map J ! A arising from the inclusion of C into A, and let A1 be the
image of the map. It is an abelian subvariety of A, and if it is not the whole of A, then there
is an abelian subvariety A2 of A such that A1 A2 ! A is an isogeny (I 10.1); in particular,
A1 \ A2 is finite. As C  A1 , this implies that C \ A2 is finite. Let W D A1  A2 and take
 to be the composite of 1  nA2 W A1  A2 ! A1  A2 with A1  A2 ! A, where n > 1
is an integer prime to the characteristic of k . Then  1 .C / is not geometrically connected
because q. 1 C / D nA12 .A2 \ C /. This is a contradiction, and so A1 must equal A.          2

R EMARK 10.4. (a). Lemma 10.2 has the following useful restatement: let V be a variety
over a field k and let D be divisor on V such that the linear system jDj is without base
points; if the map V ! Pn defined by jDj is finite, then D is ample.
    (b). If some of the major theorems from étale cohomology are assumed, then it is
possible to give a very short proof of the theorem. They show that, for any curve C on
A constructed as in the above proof, the map H 1 .A; Z` / ! H 1 .C; Z` / induced by the
inclusion of C into A is injective (see Milne 1980, VI 5.6). But H 1 .A; Z` / is dual to T` A
and H 1 .C; Z` / is dual to T` J , and so this says that the map T` J ! T` A induced by
J ! A is surjective. Clearly this implies that J maps onto A.

Q UESTION 10.5 (O PEN ). Let A be an abelian variety over an algebraically closed field k.
We have shown that there is a surjection J  A with J a Jacobian variety. Let A1 be the
subvariety of J with support the identity component of the kernel of this map. Then A1 is
an abelian variety, and so there is a surjection J1  A1 . Continuing in this way, we obtain
a sequence of abelian varieties A; A1 ; A2 ; :.. and a complex
                                    ! J2 ! J1 ! A ! 0:
Is it possible to make the constructions in such a way that the sequence terminates with 0?
That is, does there exist a finite resolution (up to isogeny) of an arbitrary abelian variety by
Jacobian varieties?


11    The zeta function of a curve
Let C be a complete nonsingular curve over a finite field k D Fq . The best way to prove the
Riemann hypothesis for C is to use intersection theory on C  C (see Hartshorne 1977, V,
118                                                        CHAPTER III. JACOBIAN VARIETIES


Ex. 1.10), but in this section we show how it can be derived from the corresponding result
for the Jacobian of C . Recall (II 1) that the characteristic polynomial of the Frobenius
endomorphism J of J acting on T` J is a polynomial P .X/ of degree 2g with integral
coefficients whose roots ai have absolute value q 1=2 .
                                                                                   P
T HEOREM 11.1. The number N of points on C with coordinates in k is equal to 1        ai C
q. Therefore, jN q 1j  2gq 1=2 .

    The proof will be based on the following analogue of the Lefschetz trace formula. A
map ˛W C ! C induces a unique endomorphism ˛ 0 of J such that f P ˛ D ˛ 0 f P for any
point P in C.k al / (cf. (6.1)).
P ROPOSITION 11.2. For any endomorphism ˛ of C ,

                                . ˛  / D 1          Tr.˛ 0 / C deg.˛/:

    Recall (I 10) that if P˛0 .X / D .X     ai /, then Tr.˛/ D ai , and that Tr.˛ 0 / D
     0
Tr.˛ jT` J /. We now show that the proposition implies the theorem. Let C W Ck al ! Ck al
be the Frobenius endomorphism of C (see II 1).
    Then . C  / D N , the degree of C is q, and the map induced by C on J is J .
Therefore the formula in (11.2) immediately gives that in (11.1). Before proving (11.2) we
need a lemma.
L EMMA 11.3. Let A be an abelian variety of dimension g over a field k, and let H be
the class of an ample divisor in NS.A/. For any endomorphism ˛ of A, write DH .˛/ D
.˛ C 1/ .H / ˛  .H / H . Then
                                                    .H g 1  DH .˛//
                                    Tr.˛/ D g                        :
                                                         .H g /

P ROOF. 5 The calculation in (I 10.13) shows that

               .˛ C n/ .H / D n.n          1/H C n.˛ C 1/ H            .n   1/˛  .H /

(because .2A / H D 4H in NS.A/), and so

                          .˛ C n/ H D n2 H C nDH .˛/ C ˛  .H /:

Now the required identity can be read off from the equation

                      P˛ . n/ D deg.˛ C n/ D ...˛ C n/ H /g /=.H g /

(AG 12.10) because P˛ . n/ D n2g C Tr.˛/n2g 1 C    .                                    2


      We now prove (11.2). Consider the commutative diagram
                                        f f                 1˛ 0
                             C C               ! J J          ! J J
                               x                    x
                               ?                    ?
                               ?                   ?
                                            f
                                C               !     J
   5 See also Kleiman, Dix Exposes, p378.
11. THE ZETA FUNCTION OF A CURVE                                                     119


where f D f P for some rational point P of C . Consider the sheaf
                                       df
                               L0 ./ D L.m          J       J  /

on J  J (see 6). Then


                     ..1  ˛ 0 /.f  f // L0 ./ D ..f  f /.1  ˛// L0 ./
                                                    ' .1  ˛/ .f  f / L0 ./
                                                    ' .1  ˛/ .LP / 1

by a formula in (6.11). Now

                       .1  ˛/ LP D L. ˛  .            P C       C  P //;

which has degree . ˛  / 1 deg.˛/. We next compute the sheaf by going round the
diagram the other way. As .1  ˛/ ı  D .1; ˛/, we have

                           ..1  ˛/ ı / L.m /  .1 C ˛/ L./;

and
                        deg f  L..1 C ˛/ .// D deg f  .1 C ˛/ :
Similarly
                          deg f  ..1  ˛// L.  J / D deg f  
and6
                         deg f  ..1  ˛// L.J  / D L.C  ˛  /;
and so we find that

                          1     . ˛  / C deg.˛/ D deg f  .D .˛//:

We know (6.12) that . g / D gŠ, and it is possible to show that f  .D .˛// D .f .C / 
D .˛// is equal to .g 1/Š. g 1  D .˛// (see Lang 1959, IV,3). Therefore (11.3)
completes the proof.

C OROLLARY 11.4. The zeta function of C is equal to

                                                       P .t/
                                     Z.C; t/ D                  :
                                                  .1   t/.1 qt/

R EMARK 11.5. As we saw in (9.6),

                              H 1 .Cet ; Z` / D H 1 .Jet ; Z` / D .T` J /_ ;

and so (11.2) can be rewritten as

                               . ˛  / D . 1/i Tr.˛jH i .Cet ; Z` //:
   6 Needs fixing.
120                                                 CHAPTER III. JACOBIAN VARIETIES


12    Torelli’s theorem: statement and applications
Torelli’s theorem says that a curve C is uniquely determined by its canonically polarized
Jacobian .J; /.
T HEOREM 12.1. Let C and C 0 be complete smooth curves over an algebraically closed
field k, and let f W C ! J and f 0 W C 0 ! J 0 be the maps of C and C 0 into their Jacobians
defined by points P and P 0 on C and C 0 . Let ˇW .J; / ! .J 0 ; 0 / be an isomorphism from
the canonically polarized Jacobian of C to that of C 0 .
  (a) There exists an isomorphism ˛W C ! C 0 such that f 0 ˛ D ˙ˇf C c, for some c in
      J 0 .k/.
  (b) Assume that C has genus  2. If C is not hyperelliptic, then the map ˛, the sign
      ˙; and c are uniquely determined by ˇ; P; P 0 . If C is hyperelliptic, the sign can be
      chosen arbitrarily, and then ˛ and c are uniquely determined.

P ROOF. (a) The proof involves complicated combinatorial arguments in the W r — we defer
it to the next section.
     (b) Recall Hartshorne 1977, IV, 5, that a curve C is hyperelliptic if there exists a finite
map W C ! P1 of degree 2; the fibres of such a map form a linear system on C of degree
2 and dimension 1, and this is the unique such linear system on C . Conversely if C has
a linear system of degree 2 and dimension 1, then the linear system defines a finite map
W C ! P1 of degree 2, and so C is hyperelliptic; the fibres of  are the members of the
linear system, and so the nontrivial automorphism  of C such that  D  preserves these
individual members.
     Now suppose that there exist ˛; ˛ 0 ; c, and c 0 such that
                                      f 0 ˛ D Cˇf C c
                                  
                                                                                             (4)
                                     f 0 ˛ 0 D Cˇf C c 0 :
Then f 0 .˛.Q// f 0 .˛ 0 .Q// D c c 0 for all Q 2 C.k/, which is a constant. Since the
fibres of the map Div0C .k/ ! J.k/ defined by f 0 are the linear equivalence classes (see
2), this implies that for all Q and Q0 in C.k/;
                             ˛.Q/     ˛ 0 .Q0 /  ˛ 0 .Q/   ˛.Q0 /,
                             ˛.Q/ C ˛ 0 .Q0 /  ˛ 0 .Q/ C ˛.Q0 /.
Suppose ˛ ¤ ˛ 0 . Then ˛.Q0 / ¤ ˛ 0 .Q0 / for some Q0 2 C.k/ and, for a suitable Q00 ,
˛.Q0 / ¤ ˛.Q00 /. Therefore j˛.Q0 / C ˛ 0 .Q00 /j is a linear system and dimension  1 (and
degree 2) on C 0 . If C (hence C 0 / is nonhyperelliptic, there is no such system, and we have
a contradiction. If C is hyperelliptic, then there is a unique linear system of dimension 1
and degree 2, but it is obvious that by varying the points Q0 and Q00 we must get more than
one system. Again we have a contradiction. We conclude that ˛ D ˛ 0 , and this implies that
c D c0.
    On the other hand, suppose that the equations (4) hold with different signs, say with a
plus and a minus respectively. Then the same argument shows that
                  ˛.Q/ C ˛ 0 .Q/  ˛.Q0 / C ˛ 0 .Q0 /, all Q; Q0 in C.k/:
Therefore f˛.Q/ C ˛ 0 .Q/ j Q 2 C.k/g is a linear system on C 0 of dimension  1, which
is impossible if C is nonhyperelliptic. (In the case C is hyperelliptic, there is an involution
 of C 0 such that ˛ D ˛ 0 :/
12. TORELLI’S THEOREM: STATEMENT AND APPLICATIONS                                           121


     The case that the equations (4) hold with minus signs can be treated the same way as
the first case.
     Finally let C 0 be hyperelliptic with an involution  such that jQ0 C Q0 j is a linear
system and f 0 .Q0 / C f 0 .Q0 / D constant. Then if f 0 ı ˛ D ˇ ı f C c, we have
f 0 ı ˛ D ˇ ı f C c 0 :                                                                  2

C OROLLARY 12.2. Let C and C 0 be curves of genus  2 over a perfect field k. If the
canonically polarized Jacobian varieties of C and C 0 are isomorphic over k, then so also
are C and C 0 .

P ROOF. Choose an isomorphism ˇW .J; / ! .J 0 ; 0 / defined over k. For each choice of a
pair of points P and P 0 in C.k al / and C 0 .k al /, there is a unique isomorphism ˛W C ! C 0
such that
                                 f P ı ˛ D ˙ˇ ı f P C c
for some c in J 0 .k al / (in the case that C is hyperelliptic, we choose the sign to be C/. Note
                                                                                     0       0
that if .P; P 0 / are replaced by the pair .Q; Q0 /, then f Q D f P C d and f Q D f P C e
for some d 2 J.k al / and e 2 J 0 .k al /, and so
            0          0
       f Q ı ˛ D f P ı ˛ C e D ˙ˇ ı f P C c C e D ˙ˇ ı f Q N
                                                          Cˇ.d / C c C e.
In particular, we see that ˛ does not depend on the choice of the pair .P; P 0 /. On applying
 2 Gal.k al =k/ to the above equation, we obtain an equation
                                    0
                               f P ı ˛ D ˙ˇ ı f P C c:
        0         0
As f P D f P and f P D f P , we see that ˛ D ˛, and so ˛ is defined over k.               2

C OROLLARY 12.3. Let k be an algebraic number field, and let S be a finite set of primes
in k. The map C 7! .JC ; / sending a curve to its canonically polarized Jacobian variety
defines an injection from the set of isomorphism classes of curves of genus  2 with good
reduction outside S into the set of isomorphism classes of principally polarized abelian
varieties over k with good reduction outside S .

P ROOF. Let R be the discrete valuation ring in k corresponding to prime of k not in S.
Then C extends to a smooth proper curve C over spec.R/, and (see 8) the Jacobian J of
C has generic fibre the Jacobian of C and special fibre the Jacobian of the reduction of C .
Therefore JC has good reduction at the prime in question. The corollary is now obvious. 2

C OROLLARY 12.4. Suppose that for any number field k, any finite set S primes of k,
and any integer g, there are only finitely many principally polarized abelian varieties of
dimension g over k having good reduction outside S . Then Mordell’s conjecture is true.

P ROOF. Combine the last corollary with (9.11).                                                2

R EMARK 12.5. Corollary (12.2) is false as stated without the condition that the genus of
C is greater than 1. It would say that all curves of genus zero over k are isomorphic to P1
(but in general there exist conics defined over k having no rational point in k/, and it would
say that all curves of genus 1 are isomorphic to their Jacobians (and, in particular, have
a rational point). However it is obviously true (without restriction on the genus) that two
curves over k having k-rational points are isomorphic over k if their canonically polarized
Jacobians are isomorphic over k.
122                                               CHAPTER III. JACOBIAN VARIETIES


13       Torelli’s theorem: the proof
        The proof that follows is short and elementary but unilluminating (to me, at least).
         There are many proofs of Torelli’s theorem, but I don’t know if there is one that is
short, elementary, and conceptual. Advice appreciated.

    Throughout this section, C will be a complete nonsingular curve of genus g  2 over
an algebraically closed field k, and P will be a closed point of C . The maps f P W C ! J
and f .r/ W C .r/ ! J corresponding to P will all be denoted by f . Therefore f .D C D 0 / D
f .D/ C f .D 0 /, and if f .D/ D f .D 0 /, then D  D 0 C rP where r D deg.D/ deg.D 0 /.
As usual, the image of C .r/ in J is denoted by W r . A canonical divisor K on C defines a
point on C .2g 2/ whose image in J will be denoted by . For any subvariety Z of J , Z 
will denote the image of Z under the map x 7!  x.
L EMMA 13.1. For all a in J.k/, .Wag 1 / D W ga 1 .

P ROOF. For any effective divisor D of degree g        1 on C ,
                        h0 .K    D/ D h1 .K      D/ D h0 .D/  1;
and so there exists an effective divisor D 0 such that K D  D 0 . Then  f .D/ a D
f .D 0 / — a, which shows that .Wag 1 /  W ga 1 . On replacing a by a, we get that
.W ga 1 /  Wag 1 , and so W ga 1 D .W ga 1 /  .W ga 1 / .                   2


L EMMA 13.2. For any r such that 0  r  g        1;
                            War  Wbg 1 ” a 2 Wbg 1 r :

P ROOF. (HW If c D f .D/Ca with D an effective divisor of degree r, and a D f .D 0 /Cb
with D 0 an effective divisor of degree g 1 r, then c D f .D C D 0 / C b with D C D 0
an effective divisor of degree g 1.
    H)W As a 2 Wbg 1 , there is an effective divisor A of degree g 1 such that a D
f .A/Cb. Let D be effective of degree r. The hypothesis states that f .D/Ca D f . N
                                                                                 D/Cb
           N                                                       N
for some D effective of degree g 1, and so f .D/ C f .A/ D f .D/ and
                                    D C A  DN C rP:
Choose effective divisors A0 and DN 0 of degree g 1 such that A C A0 and DN C DN 0 are
linearly equivalent to K (cf. the proof of 13.1). Then
                                DCK      A0  K        DN 0 C rP
and so
                                   D C DN 0  A0 C rP:
As the Ds form a family of dimension r, this shows that h0 .A0 C rP /  r C 1. (In more
detail, jA0 C rP j can be regarded as a closed subvariety of C .rCg 1/ , and we have shown
that it projects onto the whole of C .r/ .) It follows from the Riemann-Roch theorem that
h0 .K A0 rP /  1, and so there is an effective divisor AN of degree g 1 C r such that
                                    A0 C AN C rP  K:
Therefore AN C rP  K                                                N C b 2 W g 1 r .2
                                             N D f .A0 / and a D f .A/
                          A0  A, and so f .A/                                b
13. TORELLI’S THEOREM: THE PROOF                                                      123


L EMMA 13.3. For any r such that 0  r  g 1;
                                     \
                        Wg 1 r D       fW ga 1 j a 2 W r g and
                                     \
                     .W g 1 r / D     fWag 1 j a 2 W r g:

P ROOF. Clearly, for a fixed a in J.k/;

                      W g 1 r  W ga 1 ” Wag 1 r  W g 1 ;

and (13.2) shows that both hold if a 2 W r . Therefore
                                        \
                            Wg 1 r         fW ga 1 j a 2 W r g:

Conversely, c 2 W ga 1 ” a 2 W gc 1 , and so if c 2 W ga 1 for all a 2 W r , then
W r  W gc 1 and Wcr  W g 1 . According to (13.2), this implies that c 2 W g 1 r ,
which completes the proof the first equality. The second follows from the first and the
equation
                 \                         \
                    fWag 1 j a 2 W r g D      f.W ga 1 / j a 2 W r g
                                           \                     
                                        D      fW ga 1 j a 2 W r g :                  2



L EMMA 13.4. Let r be such that 0  r  g 2, and let a and b be points of J.k/ related
by an equation a C x D b C y with x 2 W 1 and y 2 W g 1 r . If WarC1 * Wbg 1 , then
WarC1 \ Wbg 1 D WaCx r
                       S with S D WarC1 \ .Wyg a2 / .

P ROOF. Write x D f .X / and y D f .Y / with X and Y effective divisors of degree 1 and
g 1 r. If Y  X , then, because f .X/ C a D f .Y / C b, we will have a D f .Y X/ C b
with Y     X an effective divisor of degree g 2 r. Therefore a 2 Wbg 2 r , and so
WarC1  Wbg 1 (by 13.2). Consequently, we may assume that X is not a point of Y .
    Let c 2 WarC1 \ Wbg 1 . Then c D f .D/ C a D f .D 0 / C b for some effective divisors
D and D 0 of degree r C 1 and g 1. Note that

                     f .D/ C y D f .D/ C a C x        b D f .D 0 / C x;

and so D C Y  D 0 C X.
     If D C Y D D 0 C X , then D  X, and so c D f .D/ C a D f .D X/ C x C a; in
                 r
this case c 2 WaCx  .
     If D C Y ¤ D C X, then h0 .D C Y /  2, and so for any point Q of C.k/, h0 .D C
                    0

Y Q/  1, and there is an effective divisor QN of degree g 1 such that D C Y  Q C Q.   N
Then
                         c D f .D/ C a D f .Q/N C a y C f .Q/;

and so c 2 fWag yCd   1
                         jd 2 W 1 g D .W g 2 /a y (by 13.3). As .W g 2 /a y D .Wyg a2 /
            T

and c is in WarC1 by assumption, this completes the proof that WarC1 Wbg 1  WaCx  r
                                                                     P
                                                                                       S.
                                                                  r         rC1   r
     The reverse inclusion follows from the obvious inclusions: WaCx  Wa ; WaCx D
   r
WbCy     Wbg 1 ; .Wyg a2 /  .Wyg a1 x / D Wbg 1 .                                    2
124                                                CHAPTER III. JACOBIAN VARIETIES


L EMMA 13.5. Let a 2 J.k/ be such that W 1 * Wag 1 ; then there is a unique effective
divisor D.a/ of degree g on C such that
                                      f .D.a// D a C                                       (5)
and W 1  Wag 1 , when regarded as a divisor on C , equals D.a/.

P ROOF. We use the notations of 6; in particular,  D W g 1 . For a D 0, (13.1) says that
                                                                                        df
. / D . Therefore, on applying (6.8), we find that W 1  Wag 1 D f .C /  . /aC D
f 1 .. /aC / D D, where D is a divisor of degree g on C such that f .g/ .D/ D a C .
This is the required result.                                                            2


    We are now ready to prove (12.1a). We use ˇ to identify J with J 0 , and write V r for
the images of C 0.r/ in J . As W g 1 and V g 1 define the same polarization of J , they give
the same element of NS.J / (see I, 10), and therefore one is a translate of the other, say
W g 1 D Vcg 1 , c 2 J.k/. To prove (12.1a), we shall show that V 1 is a translate of W 1 or
of .W 1 / .
    Let r be the smallest integer such that V 1 is contained in a translate of W rC1 or
.W  rC1 / . The theorem will be proved if we can show that r D 0. (Clearly, r < g 1:/
Assume on the contrary that r > 0. We may suppose (after possibly replacing ˇ by ˇ/
that V 1  WarC1 . Choose an x in W 1 and a y in W g 1 r , and set b D a C x y. Then,
unless WarC1  Wbg 1 , we have (with the notations of 13.4)

           V 1 \ Wbg 1 D V 1 \ WarC1 \ Wbg 1 D .V 1 \ WaCx
                                                       r
                                                           / [ .V 1 \ S/:
                           r
Note that, for a fixed a, WaCx depends only on x and S depends only on y.
   Fix an x; we shall show that for almost all y, V 1 * Wbg 1 , which implies that WarC1 *
Wbg 1 for the same y. As y runs over W g 1 r , b runs over W g.aCx/    1 r
                                                                           . Now, if V 1 
Wbg 1 for all   b in W g.aCx/
                          1 r               r
                              , then V 1  WaCx (by 13.3). This contradicts the definition
of r, and so there exist b for which V 1 * Wbg 1 . Note that V 1  Wbg 1 .D VbCc
                                                                             g 1
                                                                                 / ”
 b 2 Vcg 2 (by 13.2). Therefore Vcg 2 * W g.aCx/
                                             1 r
                                                 , and so the intersection of these sets is
a lower dimensional subset of W g.aCx/
                                   1 r
                                       whose points are the b for which V 1  Wbg 1 .
    We now return to the consideration of the intersection V 1 \ Wbg 1 , which equals .V 1 \
  r                                                                   r
WaCx / [ .V 1 \ S / for almost all y. We first show that V 1 \ WaCx        contains at most
                                                                 g 1 r
one point. If not, then as b runs over almost all points of W .aCx/ (for a fixed x/, the
                df
element D 0 .b/ D f 0 1 .V 0 Wbg 1 / (cf. 13.5) will contain at least two fixed points (because
    r          g 1           g 1
WaCx       WaCx   y D Wb        /, and hence f .D 0 .b// will lie in a translate of V g 2 . As
f 0 .D 0 .b// D b C  0 , we would then have .W g 1 r / contained in a translate of V g 2 ,
say Vdg 2 , and so

                    fVcg u1 j u 2 Vdg 2 g 
                  \                           \
                                                 fW gu 1 j u 2 .W g 1 r / g:

On applying (13.3) to each side, we then get an inclusion of V in translate of .W r / ,
contradicting the definition of r.
                                                                     r
    Keeping y fixed and varying x, we see from (5) that V 1 \ WaCx       must contain at least
one point, and hence it contains exactly one point; according to the preceding argument, the
point occurs in D 0 .b/ with multiplicity one for almost all choices of y.
14. BIBLIOGRAPHIC NOTES                                                                              125


     It is now easily seen that we can find x; x 0 in W 1 and y in W g 1 r such that .D 0 .b/ D
/D 0 .a C x y/ D Q C DN and .D 0 .b 0 / D/D 0 .a C x 0 y/ D Q0 C DN where Q; Q0 are in
C 0 and DN is an effective divisor of degree g 1 on C 0 not containing Q or Q0 . By equation
(5), f .Q/ f .Q0 / D x x 0 , and hence W 1 has two distinct points in common with some
translate of V 1 . Now, if x; x 0 are in W 1 , then W gx 1 \ W gx 0 1 D W g 2 [ .WxCx
                                                                                    g 2 
                                                                                       0 / (by

13.4). According to (13.3), we now get an inclusion of some translate of V    g  2 in W g 2 or
.W  g   2  
          / . Finally (13.3) shows that
                                          \
                                   V1 D       fV e j e 2 V g 2 g

which is contained in a translate of W 1 or W 1 according as V g 2 is contained in a trans-
late of W g 2 or .W g 2 / . This completes the proof.


14     Bibliographic notes for Abelian Varieties and Jacobian
       Varieties
[These notes will be expanded, and distributed among the various sections.]
     The theory of abelian varieties over C has a long history. On the other hand, the “abstract” theory
over arbitrary fields, can be said to have begun with Weil’s famous announcement of the proof of
the Riemann hypothesis for function fields [Sur les fonctions algébriques a corps de constantes fini,
C.R. 210 (1940) 592-594]. Parts of the projected proof (for example, the key “lemme important”)
can best be understood in terms of intersection theory on the Jacobian variety of the curve, and
Weil was to spend the next six years developing the foundational material necessary for making his
proof rigorous. Unable in 1941 to construct the Jacobian as a projective variety, Weil was led to
introduce the notion of an abstract variety (that is, a variety that is not quasi-projective). He then had
to develop the theory of such varieties, and he was forced to develop his intersection theory by local
methods (rather than the projective methods used by van der Waerden [Einfuhring in die algebraische
Geometrie, Springer, 1939]). In 1944 Weil completed his book [Foundations of algebraic geometry,
AMS Coll., XXIX, 1946], which laid the necessary foundations in algebraic geometry, and in 1946
he completed his two books [Sur les courbes algébriques et les variétés qui s’en déduisent, Hermann,
1948] and Weil 1948b, which developed the basic theory of Abelian varieties and Jacobian varieties
and gave a detailed account of his proof of the Riemann hypothesis. In the last work, abelian varieties
are defined much as we defined them and Jacobian varieties are constructed, but it was not shown
that the Jacobian could be defined over the same field as the curve.
     Chow ([Algebraic systems of positive cycles in an algebraic variety, Amer. J. Math. 72 (1950)
247-283] and Chow 1954) gave a construction of the Jacobian variety which realized it as a projec-
tive variety defined over the same ground field as the original curve. Matsusaka [On the algebraic
construction of the Picard variety, Japan J. Math 21 (1951) 217-235 and 22 (1952) 51-62] gave the
first algebraic construction of the Picard and Albanese varieties and demonstrated also that they
were projective and had the same field of definition as the original varieties. Weil showed that his
construction of a group variety starting from a birational group could also be carried out without
making an extension of the ground field [On algebraic groups of transformations, Amer. J. Math.,
77 (1955) 355-391], and in [The field of definition of a variety, Amer. J. Math., 78 (1956) 509-524]
he further developed his methods of descending the field of definition of a variety. Finally Barsotti
[A note on abelian varieties, Rend. Circ. Mat. di Palermo, 2 (1953) 236-257], Matsusaka [Some
theorems on abelian varieties, Nat. Sci. Report Ochanomizu Univ. 4 (1953) 22-35], and Weil [On
the projective embedding of abelian varieties, in Algebraic geometry and topology, A symposium in
honor of S.Lefschetz, Princeton, 1957, pp177-181] showed that all abelian varieties are projective.
In a course at the University of Chicago, 1954-55, Weil made substantial improvements to the theory
of abelian varieties (the seesaw principle and the theorem of the cube, for example), and these and
the results mentioned above together with Chow’s theory of the “k-image” and “k-trace” [Abelian
varieties over function fields, Trans. AMS, 78 (1955) 253-275] were incorporated by Lang in his
126                                                    CHAPTER III. JACOBIAN VARIETIES


book Lang 1959. The main lacuna at this time (1958/1959) was a satisfactory theory of isogenies of
degree p and their kernels in characteristic p; for example, it was not known that the canonical map
from an abelian variety to the dual of its dual was an isomorphism (its degree might have been divis-
ible by p/. Cartier [Isogenies and duality of abelian varieties, Ann of Math. 71 (1960) 315-351] and
Nishi [The Frobenius theorem and the duality theorem on an abelian variety, Mem. Coll. Sc. Kyoto
(A), 32 (1959) 333-350] settled this particular point, but the full understanding of the p-structure of
abelian varieties required the development of the theories of finite group schemes and Barsotti-Tate
groups. The book of Mumford Mumford 1970 represents a substantial contribution to the subject
of abelian varieties: it uses modern methods to give a comprehensive account of abelian varieties
including the p-theory in characteristic p, and avoids the crutch of using Jacobians to prove results
about general abelian varieties. (It has been a significant loss to the mathematical community that
Mumford did not go on to write a second volume on topics suggested in the introduction: Jacobians;
Abelian schemes: deformation theory and moduli; The ring of modular forms and the global struc-
ture of the moduli space; The Dieudonné theory of the “fine” characteristic p structure; Arithmetic
theory: abelian schemes over local, global fields. We still lack satisfactory accounts of some of these
topics.)
     Much of the present two articles has been based on these sources; we now give some other
sources and references. Abelian Varieties will be abbreviated by AV and Jacobian Varieties by JV.
     The proof that abelian varieties are projective in AV 7 is Weil’s 1957 proof. The term “isogeny”
was invented by Weil: previously, “isomorphism” had frequently been used in the same situation.
The fact that the kernel of mA has m2g elements when m is prime to the characteristic was one of the
main results that Weil had to check in order to give substance to his proof of the Riemann hypothesis.
Proposition 11.3 of AV is mentioned briefly by Weil in [Variétés abéliennes. Colloque d’algebre et
theorie des nombres, Paris, 1949, 125-128], and is treated in detail by Barsotti [Structure theorems
for group varieties, Annali di Mat. 38 (1955) 77-119]. Theorem 14.1 is folklore: it was used by Tate
in [End omorphisms of abelian varieties over finite fields, Invent. math., 2 (1966) 134-144], which
was one of the starting points for the work that led to Faltings’s recent proof of Mordell’s theorem.
The étale cohomology of an abelian variety is known to everyone who knows étale cohomology,
but I was surprised not to be able to find an adequate reference for its calculation: in Kleiman
[Algebraic cycles and the Weil conjectures, in Dix exposés sur la cohomologie des schémas, North-
Holland, 1968, pp 359-386] Jacobians are used, and it was omitted from Milne 1980. In his 1940
announcement, Weil gives a definition of the em -pairing (in our terminology, eNm -pairing) for divisor
classes of degree zero and order m on a curve which is analogous to the explicit description at the
start of 16 of AV. The results of that section mainly go back to Weil’s 1948 monograph Weil 1948b,
but they were reworked and extended to the p-part in Mumford’s book. The observation (see 16.12
of AV) that .A  A_ /4 is always principally polarized is due to Zarhin [A finiteness theorem for
unpolarized Abelian varieties over number fields with prescribed places of bad reduction, Invent.
math. 79 (1985) 309-321]. Theorem 18.1 of AV was proved by Narasimhan and Nori [Polarizations
on an abelian variety, in Geometry and Analysis, Springer, (1981), p125-128]. Proposition 20.1
of AV is due to Grothendieck (cf. Mumford [Geometric Invariant Theory, Springer, 1965, 6.1]),
and (20.5) of AV (defining the K=k-trace) is due to Chow (reference above). The Mordell-Weil
Theorem was proved by Mordell [On the rational solutions of the indeterminate equations of the
third and fourth degrees, Proc. Cambridge Phil. Soc. 21 (1922) 179-192] (the same paper in
which he stated his famous conjecture) for an elliptic curve over the rational numbers and by Weil
ŒL’arithmétique sur les courbes algébriques, Acta Math. 52 (1928) 281-315] for the Jacobian variety
of a curve over a number field. (Weil, of course, stated the result in terms of divisors on a curve.)
     The first seven sections of JV were pieced together from two disparate sources, Lang’s book
Lang 1959 and Grothendieck’s Bourbaki talks Grothendieck 62, with some help from Serre 1959,
Mumford 1966, and the first section of Katz and Mazur [Arithmetic Moduli of Elliptic Surfaces,
Princeton, 1985].
     Rosenlicht [Generalized Jacobian varieties, Ann. of Math.,59 (1954) 505-530, and A universal
mapping property of generalized Jacobians, ibid, (1957), 80-88], was the first to construct the gen-
eralized Jacobian of a curve relative to a modulus. The proof that all abelian coverings of a curve
can be obtained from isogenies of its generalized Jacobians (Theorem 9.7 of JV) is due to Lang [Sur
14. BIBLIOGRAPHIC NOTES                                                                            127


les séries L d’une variété algébrique, Bull. SMF, 84 (1956) 555-563]. Results close to Theorem 8.1
of JV were obtained by Igusa [Fibre systems of Jacobian varieties I,II,III, Amer. J. Math., 78 (1956)
p171-199, p745-760, and 81 (1959) p453-476]. Theorem 9.11 is due to Parshin [Algebraic curves
over function fields, I, Math. USSR — Izvestija, 2 (1968) 1145-1169]. Matsusaka [On a generating
curve of an abelian variety, Nat Sc. Rep. Ochanomizu Univ. 3 (1952) 1-4] showed that every abelian
variety over an algebraically closed field is generated by a curve (cf. 10.1 of JV). Regarding (11.2) of
JV, Hurwitz [Math. Ann. 28 (1886)] was the first to show the relation between the number of fixed
points of a correspondence on a Rieman surface C and the trace of a matrix describing its action
on the homology of the surface (equivalently that of its Jacobian). This result of Hurwitz inspired
both Lefschetz in his proof of his trace formula and Weil in his proof of the Riemann hypothesis for
curves.
     Proofs of Torelli’s theorem can be found in Andreotti [On a theorem of Torelli, Amer. J. Math.,
80 (1958) 801-821], Matsusaka [On a theorem of Torelli, Amer. J. Math., 80 (1958) 784-800], Weil
[Zum Beweis des Torellischen Satzes, Gott. Nachr. 2 (1957) 33-53], and Ciliberto [On a proof of
Torelli’s theorem, in Algebraic geometry — open problems, Lecture notes in math. 997, Springer,
1983 pp113-223]. The proof in 13 of JV is taken from Martens [A new proof of Torelli’s theorem,
Ann. Math. 78 (1963) 107-111]. Torelli’s original paper is [Sulle varieta di Jacobi, Rend. R. Acad.
Sci. Torino, 50 (1914-15) 439-455]. Torelli’s theorem shows that the map from the moduli space
of curves into that of principally polarized abelian varieties is injective on geometric points; a finer
discussion of the map can be found in the paper by Oort and Steenbrink [The local Torelli problem
for algebraic curves, in Algebraic Geometry Angers 1979, Sijthoff & Noordhoff, 1980, pp157-204].
     Finally, we mention that Mumford [Curves and their Jacobians, U. of Mich, Ann Arbor] pro-
vides a useful survey of the topics in its title, and that the commentaries in Weil [Collected Papers,
Springer, 1979] give a fascinating insight into the origins of parts of the subject of arithmetic geom-
etry.
     1. Artin, M: Néron models, this volume.
     2. Bourbaki, N: Algebre Commutative, Hermann, Paris (1961, 1964, 1965).
     3 Chow, W.-L: The Jacobian variety of an algebraic curve, Amer. J. Math., 76 (1954) 453-476.
     4. Grothendieck, A: Technique de descente et théoremes d’existence en géometrie algébrique, I
— VI . Séminaire Bourbaki 190, 195, 212, 221, 232, 236 (1959/62).
     5. —: Revetements étale et groupe fondamental (SGA1, 1960-61), Lecture Notes in Mathemat-
ics 224, Springer, Heidelberg (1971).
     6. —: Le groupe de Brauer, III. In Dix Exposés sur la Cohomologie des Schémas, North-
Holland, Amsterdam (1968), p88-188.
     7. — and Dieudonné, J: Eléments de géométrie algébrique I, Springer, Heidelberg, (1971).
     8. Hartshorne, R.: Algebraic Geometry, Springer, Heidelberg (1977).
     9 Lang, S.: Abelian Varieties, Interscience, New York (1959).
     10 —: Fundamentals of Diophantine Geometry, Springer, Heidelberg (1983).
     11 Lichtenbaum, S.: Duality theorems for curves over p-adic fields, Invent. math., 7 (1969)
120-136.
     12 Mattuck, A and Mayer, A. The Riemann-Roch theorem for algebraic curves, Annali Sc.
Norm. Pisa, 17, (1963) 223-237.
     13 Milne, J.: Etale Cohomology, Princeton U. P., Princeton (1980).
     14 — : Abelian varieties, this volume.
     15 Mumford, D.: Lectures on Curves on an Algebraic Surface, Princeton , 1966
     16 — : Abelian Varieties, Oxford U. P., Oxford (1970).
     17 Serre, J.-P.: Groupes Algébriques et Corps de Classes, Hermann, Paris (1959).
     18 Shafarevich, I.: Basic Algebraic Geometry, Springer, Heidelberg (1974).
     19 Waterhouse, W. Introduction to Affine Group Schemes, Springer, Heidelberg (1979).
     20 Weil, A.: Variétés abéliennes et courbes algébriques, Hermann, Paris (1948).
Chapter IV

Finiteness Theorems

At the end of the paper1 in which he proved that all the rational points on elliptic curve can
be obtained from a finite number by the tangent and chord contruction, Mordell made the
following remark:
           In conclusion, I might note that the preceding work suggests to me the
       truth of the following statements concerning indeterminate equations, none of
       which, however, I can prove. The left-hand sides are supposed to have no
       squared factors in x, the curves represented by the equations are not degenerate,
       and the genus of the equations is supposed not less than one.
           ....................
           (3) The equation

                               ax 6 C bx 5 y C : : : f xy 5 C gy 6 D z 2

       can be satisfied by only a finite number of rational values of x and y with the
       obvious extension to equations of higher degree.
          (4) The same theorem holds for the equation

                    ax 4 C by 4 C cz 4 C 2f y 2 z 2 C 2gz 2 x 2 C 2hx 2 y 2 D 0:

           (5) The same theorem holds for any homogeneous equation of genus greater
       than unity, say, f .x; y; z/ D 0.
    Statement (5) became known as Mordell’s conjecture. In this part of the course, we
discuss Faltings’s famous paper which, among other things, proves Mordell’s conjecture.
In the years since it was published, there have been some improvements and simplifications.
    Throughout, “(algebraic) number field” will mean a finite extension of Q.


1    Introduction
Mordell’s conjecture. It states:
           if C is a projective nonsingular curve of genus g  2 over a number field
       k, then C.k/ is finite.
    1 Mordell, L.J., On the rational solutions of the indeterminate equations of third and fourth degrees, Proc.

Camb. Philos. Soc. 21 (1922), 179–192.


                                                     129
130                                                      CHAPTER IV. FINITENESS THEOREMS


This was proved by Faltings in May/June 1983:
     Clearly we can omit the “projective” — removing points only makes C.k/ smaller —
and we can omit the “nonsingular” because the map C 0 ! C from the desingularization
(normalization) C 0 of C to C induces a map C 0 .k/ ! C.k/ that becomes bijective when a
finite number of points are removed from C 0 .k/ and C.k/. However, one must be careful
to check that the genus of the associated complete nonsingular curve is  2.
     We illustrate this by examining when Faltings’s theorem implies that the equation
                                     X
                    F .X; Y; Z/ D           aij k X i Y j Z k D 0, aij k 2 k
                                        i Cj CkDn

has only finitely many solutions in k (counted in the sense of projective geometry).
    First we need to assume that F .X; Y; Z/ is absolutely irreducible, i.e., that it is irre-
ducible and remains so over every extension of k. This is not a serious restriction, because
F .X; Y; Z/ will factor into absolutely irreducible polynomials over a finite extension k 0 of
k, and we can replace F with one of the factors and k with k 0 . Thus, we may suppose that
F .X; Y; Z/ defines a complete geometrically irreducible curve over k. The genus of the
associated nonsingular curve is

                                           .n     1/.n     2/
                                     gD                          ˙ nP
                                                   2
(Plücker’s formula)2 where the sum is over the singular points on the curve F .X; Y; Z/ D
0 with coordinates in C. There are formulas for nP . For example, if P is an ordinary
singularity with multiplicity m, then

                                          nP D m.m            1/=2:

    If g  2, then Faltings’s theorem states that C.k/ is finite. For example, the Fermat
curve
                                X n C Y n D Z n , n  4;
has only finitely many solutions in any number field (up to multiplication by a constant).
    If g D 1, then either C.k/ is empty or there is a map from a finitely generated abelian
group to C.k/ that becomes bijectivepwhen a finite number of points are removed from each
of the curves. For example, over QŒ 3 D for certain D, the points on

                                             X 3 C Y 3 D Z3

form an abelian group of rank  3.
    If g D 0, then either C.k/ is empty, or there is a map P1 .k/ ! C.k/ that becomes
bijective when a finite number of points are removed from each of the curves. For example,
for the curve
                                      X 2 C Y 2 D Z2
there is a bijection

                     P1 .k/ ! C.k/;         .t W u/ 7! .t 2     u2 W 2tu W t 2 C u2 /:
    2 See Fulton, W., Algebraic Curves, Benjamin, 1969, p199 for a proof of Plücker’s formula in the case that

C has only ordinary singularities.
1. INTRODUCTION                                                                             131


There is an algorithm for deciding whether a curve of genus 0 over Q has a rational point.
Thus, except for g D 1, we have an algorithm for deciding whether C.k/ is finite —
therefore, g D 1 is the interesting case!
     It is possible to give a bound for #C.k/ — this is not entirely clear from Faltings’s
approach, but it is clear from the Vojta-Faltings-Bombieri approach. However, there is at
present no algorithm for finding all the points on C . For this, one would need an effective
bound on the heights of the points on C (for a point P D .x W y W z/ 2 P2 .Q/; H.P / D
max.jxj; jyj; jzj/ where x; y; z are chosen to be relatively prime integers). With such a
bound N , one would only need to check whether each of the finitely many points P with
H.P /  N lies on C . Finding an effective bound on the heights appears to be an extremely
difficult problem: for example, it was only in the 1960s that Baker showed that there was a
bound on the heights of the integer solutions of Y 2 D X 3 C k (for which he received the
Fields medal).
     Heuristic argument for the conjecture. Let C be a complete nonsingular curve over a
number field k, and let J be its Jacobian variety. If C.k/ is empty, then it is certainly finite.
Otherwise there is an embedding C ,! J . Consider the diagram:

                                      C.C/         ,! J.C/
                                        "              "
                                C.k/ D C.C/ \ J.k/ ,! J.k/

According to the Mordell-Weil theorem, J.k/ is a finitely generated group, and if g  2,
then
                                dim C.C/ < dim J.C/:
Since there is no reason to expect any relation between C.C/ and J.k/ as subsets of J.C/,
and both are sparse, C.C/ \ J.k/ should be finite. People have tried to make this into a
proof, but without success3 .
    Finiteness I and its Consequences. Most of the main theorems of Faltings’s paper
follow from the following elementary statement.
T HEOREM 1.1 (F INITENESS I). Let A be an abelian variety over an algebraic number field
k. Then, up to isomorphism, there are only finitely many abelian varieties B over k that are
isogenous to A.

    In other words, the abelian varieties over k isogenous to A fall into only finitely many
isomorphism classes. At first sight, this statement is rather surprising. Let ˛W A ! B be
                       df
an isogeny. Then N D Ker.˛/ is a finite subgroup variety of A, and N.k al / is a finite
subgroup of A.k al / stable under the action of . Conversely, from every finite subgroup
N of A.k al / stable under Gal.k al =k/ we get an isogeny A ! A=N . Clearly, there are
infinitely many possible N ’s, but of course there may be isomorphisms A=N  A=N 0 ; for
                                       n
example, A  A=An , An D Ker.A ! A/. The theorem is a rather strong statement about
the absence of exotic finite subgroups of A.k al / stable under Gal.k al =k/, and about the
existence of isomorphisms between the quotients A=N .
    Finiteness I implies the following theorems:
T HEOREM 1.2 (S EMISIMPLICITY ). Let A be an abelian variety over a number field k; for
all primes `, the action of Gal.k al =k/ on V` A is semisimple.
   3 There has been progress on these questions since the notes were written.
132                                             CHAPTER IV. FINITENESS THEOREMS


T HEOREM 1.3 (TATE ’ S CONJECTURE ). For abelian varieties A and B over a number field
k, the map

                Hom.A; B/ ˝ Z` ! Hom.T` A; T` B/ ,            D Gal.k al =k/;

is bijective.

T HEOREM 1.4 (F INITENESS II). Given a number field k, an integer g, and a finite set of
finite primes S of k, there are only finitely many isomorphism classes of abelian varieties
A over k of dimension g having good reduction outside S .

     For elliptic curves, Finiteness II was proved by Shafarevich — see Silverman 1986, IX,
Theorem 6.1. Faltings’s proof is (necessarily) completely different.
     That V` A is a semisimple -module means that every subspace W of V` A stable under
the action of has a complement W 0 also stable under , i.e., V` A D W ˚ W 0 with W 0
  -stable. This implies that V` A is a direct sum of simple Q` Œ -modules (i.e., subspaces
stable under with no nontrivial -stable subspaces).
     The action of a finite group on a finite-dimensional vector space over a field of charac-
teristic zero is automatically semisimple (see 10.2). Essentially the same proof as in (10.2)
shows that the actionPof a compact  R group on a finite-dimensional vector space over R is
semisimple (replace g with g ). However, this is not true for a compact group act-
ing on a finite-dimensional vector space over Q` . For example the action of the compact
group                             ˚      
                                D a0 bc j ac D 1; a; b; c 2 Z`
on Q2` is not semisimple because f. 0 /g is a -stable subspace having no -stable comple-
ment.
    The Tate conjecture has been discussed already in (10.17). Faltings’s methods also
allow one to prove it for a field k finitely generated over Q. It was known (Zarhin, Izv.
1975) that Finiteness II implies the Tate conjecture. Faltings turned things around by
    (i) proving a weak form of Finiteness II;
    (ii) proving the Tate conjecture;
    (iii) deducing Finiteness II.
Finiteness II implies the following result:

T HEOREM 1.5 (S HAFAREVICH ’ S CONJECTURE ). Given a number field k, an integer g,
and a finite set of finite primes S of k, there are only finitely many isomorphism classes of
nonsingular complete curves C over k of genus g having good reduction outside S.

   This is proved by applying Finiteness II to the Jacobians of the curves (see later).
   In 1968, Parshin showed that Shafarevich’s conjecture implies Mordell’s conjecture.
The idea of the proof is to attach to a point P in C.k/ a covering

                                       ' P W CP ! Ck 0

where
  (a) .CP ; 'P / is defined over a fixed finite extension k 0 of k;
  (b) CP has bounded genus,
  (c) CP has good reduction outside a fixed finite set of primes of k 0 ;
1. INTRODUCTION                                                                              133


  (d) 'P is ramified exactly at P .
    The statements (a),(b),(c) and Shafarevich’s conjecture show that there are only finitely
many curves CP , and (d) shows that the map P 7! .CP ; 'P / is injective. Finally, a classical
theorem of de Franchis states that, for fixed C 0 and C , there can be only finitely many
surjective maps C 0 ! C when C has genus  2, and so P 7! CP is finite-to-one. (This is
the only place in the argument that g  2 is used!)
    The proof of Finiteness I. Here I briefly sketch the proof of Finiteness I. In the next
section, we define the notion of semistable reduction for an abelian variety (it is weaker
than good reduction), and we note that an abelian variety acquires semistable reduction at
every prime after a finite extension of the ground field.
    Given an abelian variety A over a number field, Faltings attaches a real number, h.A/ to
A, called the Faltings height of A. The Faltings heights of two isogenous abelian varieties
are related, and Faltings proved:
T HEOREM 1.6. Let A be an abelian variety with semistable reduction over a number field
k. The set
                           fh.B/j B is isogenous to Ag
is finite.

    There is natural notion of the height of a point in Pn .k/, namely, if P D .a0 W    W an /,
then                                       Y
                                 H.P / D         max.jai jv /:
                                                  v   i
Here the v’s run through all primes of k (including the archimedean primes) and jjv denotes
the normalized valuation corresponding to v. Note that
         Y                       Y                   Y             Y
               max.jcai jv / D .       max.jai jv //.     jcjv / D     max.jai jv /
             v    i                   v   i               v              v   i
                                              Q
because the product formula shows that v jcjv D 1. Therefore H.P / is independent of
the choice of a representative for P . When k D Q , we can represent P by an n-tuple
.a0 W ::: W an / with the ai relatively prime integers. Then maxi .jai jp / D 1 for all prime
numbers p, and so the formula for the height becomes

                          H.P / D max jai j (usual absolute value).
                                      i

A fundamental property of heights is that, for any integer N ,

                               CardfP 2 Pn .k/ j H.P /  N g

is finite. When k D Q, this is obvious.
     Using heights on projective space, it is possible to attach another height to an abelian
variety. There is a variety V (the Siegel modular variety) over Q that parametrizes isomor-
phism classes of principally polarized abelian varieties of a fixed dimension g. It has a
canonical class of embeddings into projective space

                                              V ,! Pn :

An abelian variety A over k corresponds to a point v.A/ in V .k/, and we define the modular
height of A to be
                                   H.A/ D H.v.A//:
134                                               CHAPTER IV. FINITENESS THEOREMS


We know that the set of isomorphism classes of principally polarized abelian varieties over
k of fixed dimension and bounded modular height is finite.
    Note that if we ignore the “principally polarized” in the last statement, and the “semi-
stable” in the last theorem, then they will imply Finiteness I once we relate the two notions
of height. Both heights are “continuous” functions on the Siegel modular variety, which
has a canonical compactification. If the difference of the two functions h and H extended
to the compact variety, then it would be bounded, and we would have proved Finiteness I.
Unfortunately, the proof is not that easy, and the hardest part of Faltings’s paper is the study
of the singularities of the functions as they approach the boundary. One thing that makes
this especially difficult is that, in order to control the contributions at the finite primes, this
has to be done over Z, i.e., one has to work with a compactification of the Siegel modular
scheme over Z.

References.
The original source is:
    Faltings, G., Endlichkeitssätze für Abelsche Varietäten über Zahlkörpern, Invent. Math.
73 (1983), 349-366; Erratum, ibid. 1984, 75, p381. (There is a translation: Finiteness
Theorems for Abelian Varieties over Number Fields, in “Arithmetic Geometry” pp 9–27.)
    Mathematically, this is a wonderful paper; unfortunately, the exposition, as in all of
Faltings’s papers, is poor.
   The following books contain background material for the proof:
   Serre: Lectures on the Mordell-Weil theorem, Vieweg, 1989.
   Arithmetic Geometry (ed. Cornell and Silverman), Springer, 1986 (cited as Arithmetic
Geometry).
    There are two published seminars expanding on the paper:
    Faltings, G., Grunewald, F., Schappacher, N., Stuhler, U., and Wüstholz, G., Rational
Points (Seminar Bonn/Wuppertal 1983/84), Vieweg 1984.
    Szpiro, L., et al. Séminaire sur les Pinceaux Arithmétique: La Conjecture de Mordell,
Astérisque 127, 1985.
    Although it is sketchy in some parts, the first is the best introduction to Faltings’s paper.
In the second seminar, the proofs are very reliable and complete, and they improve many of
the results, but the seminar is very difficult to read.
      There are two Bourbaki talks:
      Szpiro, L., La Conjecture de Mordell, Séminaire Bourbaki, 1983/84.
      Deligne, P., Preuve des conjectures de Tate et Shafarevitch, ibid.
      There is a summary of part of the theory in:
      Lang, S., Number Theory III, Springer, 1991, Chapter IV.
    Faltings’s proofs depend heavily on the theory of Néron models of abelian varieties and
the compactification of Siegel modular varieties over Z. Recently books have appeared on
these two topics:
    Bosch, S., Lütkebohmert, W., and Raynaud, M., Néron Models, Springer, 1990.
    Chai, Ching-Li and Faltings, G., Degeneration of Abelian Varieties, Springer, 1990.
2. THE TATE CONJECTURE; SEMISIMPLICITY.                                                 135


2       The Tate Conjecture; Semisimplicity.
In this section, we prove that Tate’s conjecture is implied by Finiteness I. Throughout the
section, k is a field and D Gal.k al =k/. We begin with some elementary lemmas.

L EMMA 2.1. If ˛W A ! B is an isogeny of degree prime to chark, then Ker.˛/.k al / is
a finite subgroup of A.k al / stable under the action of ; conversely, every such subgroup
arises as the kernel of such an isogeny, i.e., the quotient A=N exists over k.

P ROOF. Over k al , this follows from (8.10). The only additional fact needed is that, if
N.k al / is stable under the action of , then the quotient A=N is defined over k.      2


L EMMA 2.2. (a) For any abelian variety A and ` ¤ char.k/, there is an exact sequence
                                        `n
                            0 ! T` A ! T` A ! A`n .k al / ! 0:

(b) An isogeny ˛W A ! B of degree prime to char.k/ defines an exact sequence

                                   0 ! T` A ! T` B ! C ! 0

with the order of C equal to the power of ` dividing deg.˛/..

P ROOF. (a) This follows easily from the definition

               T` A D f.an /n1 j an 2 A`n .k al /;     `an D an 1 ;   `a1 D 0g:

(b) To prove this, consider the following infinite diagram:
                                    x                    x
                                    ?                    ?
                                    ?`                   ?`
                                                ˛
    0      !    Kn         ! B`n .k al /            !    A`n .k al     !   Cn      ! 0
                x               x                           x              x
                ?               ?                           ?              ?
                ?`              ?`                          ?`             ?
                                                ˛
    0      ! KnC1          ! B`nC1 .k al /          ! A`nC1 .k al /    ! CnC1      ! 0
                                 x                        x
                                 ?                        ?
                                 ?`                       ?`

For n sufficiently large, Kn D KnC1 D : : : D K, say. Because K is finite, it has no
element divisible by all powers of `, and so
                              df
                     lim Kn D f.an /j an 2 Kn , `an D an 1 , `a1 D 0g

is zero. Since #B`n .k al / D .`n /2g D #A`n .k al /, we must have #Kn D #Cn . Therefore
#Cn is constant for n large. The map CnC1 ! Cn is surjective; therefore for n large it is
bijective, and it follows that lim Cm ! Cn is a bijection for all large n. On passing to the
inverse limit we get an exact sequence

                                   0 ! T` B ! T` A ! C ! 0

as required.                                                                              2
136                                                    CHAPTER IV. FINITENESS THEOREMS


   Let ˛W B ! A be an isogeny. Then the image of T` ˛W T` B ! T` A is -stable Z` -
module of finite index in T` A. Our final elementary lemma shows that every such sub-
module arises from an isogeny ˛, and even that ˛ can be taken to have degree a power of
`.

L EMMA 2.3. Assume ` ¤ char.k/. For any -stable submodule W of finite index in T` A,
there an abelian variety B and an isogeny ˛W B ! A of degree a power of ` such that
˛.T` B/ D W .

P ROOF. Choose n so large that W  `n T` A, and let N be the image of W in T` A=`n T` A D
A`n .k al /. Then N is stable under the action of , and we define B D A=N . Because
N  A`n , the map `n W A ! A factors through A ! A=N :

                                                  `n              /: A
                                A JJ
                                     JJ
                                       JJ                      uuu
                                                              u
                                         JJ                 uu
                                       ˇ JJ$              uu
                                                        uu ˛
                                           A=N D B

It remains to show that I m.T` ˛W T` B ! T` A/ D W . From the diagram

                                                  `n          / T` A
                                 T` AE
                                        EE                   yy<
                                           EE              yy
                                             E           yyT` ˛
                                       T` ˇ EE"        yy
                                               T` B

it is clear that I m.T` ˛/  `n T` A, and so it suffices to show that the image of I m.T` ˛/ in
T` A=`n T` A D A`n .k al / is N . But

                          B.k al /`n D fa 2 A.k al /j `n a 2 N g=N;

and if b 2 B.k al /`n is represented by a 2 A.k al /, then ˛.b/ D `n a. It is now clear that ˛
maps B.k al /`n onto N .                                                                     2


    Let A be an abelian variety over a field k, and let ` be a prime ¤ chark. Consider the
following condition (slightly weaker than Finiteness I):
          (*) up to isomorphism, there are only finitely many abelian varieties B
      isogenous to A by an isogeny of degree a power of `.

L EMMA 2.4. Suppose A satisfies (*). For any W  V` A stable under            , there is a u 2
End.A/ ˝ Q` such that uV` A D W .

P ROOF. Set T` D T` A and V` D V` A. Let

                                  Xn D .T` \ W / C `n T` :

This is a Z` -submodule of T` stable under and of finite index in T` . Therefore, there is
an isogeny
                      fn W B.n/ ! A, such that fn .T` B.n// D Xn :
2. THE TATE CONJECTURE; SEMISIMPLICITY.                                                   137


According to (*), the B.n/ fall into only finitely many distinct isomorphism classes, and so
at least one class has infinitely many B.n/’s: there is an infinite set I of positive integers
such that all the B.i / for i 2 I are isomorphic. Let i0 be the smallest element of I . For
each i 2 I , choose an isomorphism vi W B.i0 / ! B.i/, and consider:
                               vi
                     B.i0 /         ! B.i/     T` B.i0 /     ! T` B.i/
                      ?                ?          ?               ?
                                                yfi0           y fi
                      ?f                          ?               ?
                                       yfi
                                       ?
                      y i0
                       A               A         X i0              Xi
                                df
Because fi0 is an isogeny, ui D fi vi fi0 1 makes sense as an element of End.A/ ˝ Q,
and hence as an element of End.A/ ˝ Q` . Moreover, it is clear from the second diagram
that ui .Xi0 / D Xi . Because Xi  Xi0 , the ui for i 2 I are in the compact set End.Xi0 /,
and so, after possibly replacing .ui / with a subsequence, we can assume .ui / converges
to a limit u in End.Xi0 /  End.V` A/. Now End0 .A/` is a subspace of End.V` A/, and
hence is closed. Since each ui lies in End.A/ ˝ Q` , so also does their limit u. For any
x 2 Xi0 , u.x/ D lim ui .x/  \Xi . Conversely, if y 2 \Xi , then there exists for each
i 2 I , an element xi 2 Xi0 such that ui .xi / D y. From the compactness of Xi0 again, we
deduce that, after possibly replacing I with a subset, the sequence .xi / will converge to a
limit x 2 Xi0 . Now u.x/ D lim u.xi / D lim ui .xi / D y. Thus u.Xi0 / D \Xj D T` \ W ,
and it follows that u.V` A/ D W .                                                         2

     Before proving the main theorem of this section, we need to review a little of the theory
of noncommutative rings (CFT, Chapter IV). By a k-algebra, I will mean a ring R, not
necessarily commutative, containing k in its centre and of finite dimension over k, and
by an R-module I’ll mean an R-module that is of finite dimension over k. If R has a
faithful semisimple module, then every R-module is semisimple, and the k-algebra R is
said to be semisimple. A simple k-algebra, i.e., a k-algebra with no two-sided ideals except
for the obvious two, is semisimple and a theorem of Wedderburn says that, conversely, a
semisimple k-algebra is a finite product of simple k-algebras.
     Another theorem of Wedderburn says that every simple k-algebra is isomorphic to
Mn .D/ for some n and some division k-algebra D.
     Let D be a division algebra over k. The right ideals in Mn .D/ are the sets of the form
a.J / with J  f1; 2; : : : ; ng and a.J / the set of matrices whose j th columns are zero for
j … J . Note that a.J / is generated by the idempotent e D diag.a1 ; : : : ; an / with aj D 1
for j 2 J and aj D 0 otherwise. On combining this remark with the Wedderburn theorems,
we find that every right ideal in a semisimple k-algebra R is generated by an idempotent:
a D eR for some e with e 2 D e.
     The centralizer CE .R/ of subalgebra R of a k-algebra E consists of the elements of
E such that ˛ D ˛ for all ˛ 2 R. Let R be a k-algebra and let E D Endk .V / for some
faithful semisimple R-module V ; the Double Centralizer Theorem says that CE .CE .R// D
R:
     If R is a semisimple k-algebra, then R ˝k k 0 need not be semisimple — for example, if
R D kŒ˛ with ˛ p 2 k, ˛ … k, then R ˝k k al contains the nilpotent element ˛ ˝ 1 1 ˝ ˛.
However, this only happens in characteristic p: if k is of characteristic 0, then R semisimple
 H) R ˝k k 0 semisimple.
     Let A be an abelian variety. Then End.A/ ˝ Q is a finite-dimensional algebra over Q
(10.15), and it is isomorphic to a product of matrix algebras over division algebras (see the
first subsection of 9). It is, therefore, a semisimple Q-algebra.
138                                             CHAPTER IV. FINITENESS THEOREMS


T HEOREM 2.5. Let A be an abelian variety over k, and assume that A  A and A satisfy
(*) for some ` ¤ char.k/. Then
  (a) V` A is a semisimple Q` Œ -module.
  (b) End.A/ ˝ Q` D End.V` A/ .

P ROOF. (a) Let W be a -subspace of V` A — we have to construct a complement W 0 to
W that is stable under . Let
                          a D fu 2 End.A/ ˝ Q` j uV` A  W g:
This is a right ideal in End.A/˝Q` , and aV` A D W because the hypothesis on A and (2.4)
imply there exists a u 2 End.A/ ˝ Q` such that uV` A D W . From the above remarks,
we know that a is generated by an idempotent e, and clearly eV` D W . Because e is
idempotent
                           V` A D eV` A ˚ .1 e/V` A D W ˚ W 0 :
                                                                               df
Since the elements of commute with the elements of End.A/ ˝ Q` , W 0 D .1 e/V` A
is stable under the action of . (b) Let C be the centralizer of End.A/ ˝ Q` in End.V` A/,
and let B be the centralizer of C . Because End.A/˝Q` is semisimple, B D End.A/˝Q` .
Consider ˛ 2 End.V` A/ — we have to show that ˛ 2 B. The graph of ˛
                                     df
                                 W D f.x; ˛x/j x 2 V` Ag
is a   -invariant subspace of V` A  V` A, and so there is a u 2End.A  A/ ˝ Q` D
                                                                    c 0
M2 .End.A//˝Q` such that u.V` .AA// D W . Let c 2 C . Then               2 End.V` A
                                                                    0 c
V` A/ commutes with End.A  A/ ˝ Q` , and, in particular, with u. Consequently,
                                                         
                   c 0             c 0                  c 0
                          W D             uV` A D u             V` A  W:
                   0 c             0 c                  0 c
This says that, for any x 2 V` A, .cx; c˛x/ 2 W Dgraph of ˛. Thus ˛ maps cx to c˛x,
i.e., ˛cx D c˛x. Thus c˛ D ˛c, and since this holds for all c, ˛ 2 B D End.A/ ˝ Q` . 2

C OROLLARY 2.6. Assume (*) holds for abelian varieties over k. Then the map
                         Hom.A; B/ ˝ Q` ! Hom.V` A; V` B/
is an isomorphism.

P ROOF. Consider the diagram of finite-dimensional vector spaces over Q` :
       End.V` .AB// D End.V` A/  Hom.V` A;V` B/  Hom.V` B;V` A/  End.V` B/
              [            [             [                [              [
       End.AB/˝Q` D End.A/˝Q`  Hom.A;B/˝Q`  Hom.B;A/˝Q`  End.B/˝Q` :
The theorem shows that the inclusion at left is an equality, and it follows that the remaining
inclusions are also equalities.                                                              2

C OROLLARY 2.7. Let R be the image of Q` Œ  in End.V` A/. Then R is the centralizer
of End0 .A/` in End.V` A/.

P ROOF. Theorem 2.5a shows that V` A is a semisimple R-module. As it is also faithful, this
implies that R is a semisimple ring. The double centralizer theorem says that C.C.R// D
R, and (2.5b) says that C.R/ D End.A/ ˝ Q` . On putting these statements together, we
find that C.End.A/ ˝ Q` / D R:                                                           2
3. FINITENESS I IMPLIES FINITENESS II.                                                           139


3    Finiteness I implies Finiteness II.
In this section we assume Finiteness I (up to isomorphism, there are only finitely many
abelian varieties over a number field k isogenous to a fixed abelian variety). Hence we can
apply Tate’s conjecture and the semisimplicity theorem.
    We first need a result from algebraic number theory which is the analogue of the the-
orem that a compact Riemann surface has only finitely many coverings with fixed degree
unramified outside a fixed finite set.

T HEOREM 3.1. For any number field K, integer N , and finite set of primes S of K, there
are only finitely many fields L  K unramified outside S and of degree N (up to K-
isomorphism of course).

P ROOF. First recall from ANT, 7.65, that for any prime v and integer N , there are only
finitely many extensions of Kv of degree dividing N (Kv D completion of K at v). This
follows from Krasner’s lemma: roughly speaking, such an extension is described by a monic
polynomial P .T / of degree d jN with coefficients in Ov ; the set of such polynomials is
compact, and Krasner’s lemma implies that two suchQ polynomials that are close define the
same extension. Now, recall that Disc.L=K/ D Disc.Lw =Kv / (in an obvious sense),
and because we are assuming L is ramified only at primes in S, the product on the right is
over the primes w dividing a prime v in S . Therefore Disc.L=K/ is bounded, and we can
apply the the following classical result.


T HEOREM 3.2 (H ERMITE 1857). There are only finitely many number fields with a given
discriminant (up to isomorphism).

P ROOF. Recall (ANT 4.3) that, for an extension K of Q of degree n, there exists a set of
representatives for the ideal class group of K consisting of integral ideals a with

                                        nŠ 4 s
                                            
                                                             1
                               N.a/  n          jDiscK=Q j 2 :
                                        n    

Here s is the number of conjugate pairs of nonreal complex embeddings of K. Since N.a/ >
1, this implies that
                                               2s  nn 2
                              jDiscK=Q j >                   :
                                               4       nŠ
        n
Since nnŠ ! 1 as n ! 1 (by Stirling’s formula, if it isn’t obvious), we see that if we
bound jDiscK=Q j then we bound n. Thus, it remains to show that, for a fixed n, there are
only finitely many number fields with a given discriminant d . Let D D jd j. Let 1 ; : : : ; r
be the embeddings of F into R, and let rC1 ; N rC1 ; : : : ; rCs ; N rCs be the complex em-
beddings. Consider the map

             W K ! RrCs ;     x 7! .1 .x/; : : : ; r .x/; <rC1 .x/; =rC1 .x/; : : :/:

In the case that r ¤ 0, define X to be the setp
                                              of n-tuples .x1 ; : : : ; xr ; yrC1 ; zrC1 ; : : :/ such
that jxi j < Ci and yj2 Czj2 < 1, where C1 D D C 1 and Ci D 1 for i ¤ 1. In the contrary
140                                               CHAPTER IV. FINITENESS THEOREMS

                                                                                          p
case, define Y to be the set of n-tuples .y1 ; z1 ; : : :/ such that jy1 j < 1, jz1 j <       D C 1,
and yi2 C zi2 < 1 for i > 1. One checks easily that the volumes of these sets are
                                      p                           p
                      .X / D 2r  s 1 C D; .Y / D 2 s 1 1 C D;
                                    p                  p
and so both quotients .X /=2r D and .Y /= D are greater than 1. By Minkowski’s
Theorem (ANT 4.19), there exist nonzero integers in K that are mapped into X or Y ,
according to the case. Let ˛ be one of them. Since its conjugates are absolutely bounded
by a constant depending only on D, the coefficients of the minimum polynomial of ˛ over
Q are bounded, and so there are only finitely many possibilities for ˛. We shall complete
the proof by showing that K D QŒ˛. If r ¤ 0, then 1 ˛ is the only conjugate of ˛ lying
outside the unit circle (if it didn’t lie outside, then NmK=Q .˛/ < 1). If r D 0, then 1 ˛ and
N 1 ˛ are the only conjugates of ˛ with this property, and 1 ˛ ¤ N 1 ˛ since otherwise every
conjugate of ˛ would lie on the unit circle. Thus, in both cases, there exists a conjugate of
˛ that is distinct from all other conjugates, and so ˛ generates K.                           2


    Let K be a number field, and let L be a Galois extension of K with Galois group G.
Let w be a prime of L. The decomposition group is

                            D.w/ D f 2 Gal.L=K/ j w D wg:

The elements of D.w/ act continuously on L for the w-adic topology, and therefore extend
to the completion Lw of L. In fact Lw is Galois over Kv with Galois group D.w/. The
group D.w/ acts on the residue field k.w/, and so we get a homomorphism

                                  D.w/ ! Gal.k.w/=k.v//:

The kernel is called the inertia group I.w/. When I.w/ D 1, L is said to be unramified
over K at w, and we define the Frobenius element Frobw at w to be the element of D.w/
corresponding to the canonical generator of Gal.k.w/=k.v//. Thus Frobw is the unique
element of G such that

                     Frobw .Pw / D Pw ;       Frobw .a/  aqv (mod Pw /

where Pw is the prime ideal of L corresponding to w, qv D #k.w/, and a is any element
of the ring of integers of L. Because L is Galois, the decomposition groups at the primes
lying over v are conjugate, and so are the inertia groups. Therefore, if one prime w lying
over v is unramified they all are, and fFrobw j wjvg is a conjugacy class in G — we denote
it by .v; L=K/.

T HEOREM 3.3 (C HEBOTAREV DENSITY THEOREM ). Let L be a finite Galois extension
of a number field K with Galois group G. Let C be a subset of G stable under conjugation.
Then the set of primes v of L such that .v; L=K/ D C has density jC j=jGj:

P ROOF. For a discussion of the theorem, see ANT, 8.31, and for a proof, see CFT, VIII
7.                                                                                 2


R EMARK 3.4. The theorem is effective, i.e., given a class C , there is a known bound B
such that there will be a prime v with N.v/  B for which .v; L=K/ D C .
3. FINITENESS I IMPLIES FINITENESS II.                                                      141


    Now consider an infinite Galois extension L over K with Galois group G. Recall (FT,
7) that G has a natural topology for which it is compact, and that the main theorem of
Galois theory holds for infinite extension, except that it now provides a one-to-one corre-
spondence between the intermediate fields M , L  M  K, and the closed subgroups of
G. The above definitions of decomposition group etc. still make sense for infinite exten-
sions. (One difference: the set of primes ramifying in L may be infinite.)
                                                                              df
    Let V be a finite dimensional vector space over Q` . A representation of D Gal.K al =K/
on V is a continuous homomorphism

                                W    ! GL.V / Ddf Aut.V /:

The kernel of  is a closed normal subgroup of , corresponding to a (possibly infinite)
Galois extension L of K. The representation  is said to be unramified at a prime v of K if
v is unramified in L.
    We are especially interested in the representation of        on V` A, A an abelian variety
over K. Then the field L in the last paragraph is the smallest extension of K such that all
the `-power torsion points of A are rational over it, i.e., such that A.L/.`/ D A.K al /.`/.
T HEOREM 3.5. Let A be an abelian variety over a number field K. Let v be a finite prime
of K, and let ` be a prime distinct from the characteristic of k.v/ (i.e., such that v - `).
Then A has good reduction at v if and only if the representation of Gal.K al =K/ on V` A is
unramified at v.

P ROOF. ): For elliptic curves, this is proved in Silverman, 1986, VII 4.1. The proof for
abelian varieties is not much more difficult. (: For elliptic curves, see Silverman, 1986,
7.1. As we now explain, the statement for abelian varieties is an immediate consequence
of the existence of Néron models (and hence is best called the Néron criterion). Clearly the
statement is really about A regarded as an abelian variety over the local field Kv . As we
noted in 20, Néron showed that there is a canonical way to pass from an abelian variety
A over Kv to a commutative algebraic group A0 over the residue field k D k.v/. For any
prime ` ¤ char.k.v//, the reduction map

                                      A.Kv /`n ! A0 .k/`n

is a bijection. The algebraic group A0 doesn’t change when Kv is replaced by an unramified
extension. It has a filtration whose quotients are successively a finite algebraic group F (i.e.,
an algebraic group of dimension 0), an abelian variety B, a torus T , and an additive group
U . We have
                              dim A D dim B C dim T C dim U .
Moreover: #B.k al /`n D `2n dim.B/ ; #T .k al /`n D `ndim.T / , because Tk al  Gdim
                                                                                 m
                                                                                     T,

Gm .L/ D L all fields L  Q; #U.k al /`n D 0, because Uk al  Gdim   U , G .L/ D L all
                                                                   a       a
fields L  Q. Now suppose that A has good reduction, so that A0 D B. For all n,

                                     A.Kvun /`n D A0 .k al /`n

has `2n dim A elements, and so A.Kvun /`n D A.Kval /`n . Therefore the action of Gal.Kval =Kv /
on V` A factors through Gal.Kvun =Kv /, which is what it means for the representation of
Gal.Kval =Kv / on V` A to be unramified. On the other hand, if A does not have good reduc-
tion, then
                          #A.Kvun /`n D #A0 .k al /`n < `2n dim A
142                                             CHAPTER IV. FINITENESS THEOREMS


for n sufficiently large. As
                                                          al   un
                               A.Kvun /`n D A.Kval /Gal.Kv =Kv /

this shows that
                                      al   un
                        A.Kval /Gal.Kv =Kv / ¤ A.Kval /`n ;    n >> 0:
Therefore the representation of the Galois group on V` A is ramified at v.                   2


C OROLLARY 3.6. If A and B are isogenous over K, and one has good reduction at v, then
so also does the other.

P ROOF. The isogeny defines an isomorphism V` A ! V` B commuting with the actions of
Gal.K al =K).                                                                     2


     Recall that for an abelian variety A over a finite field k with q elements, the character-
istic polynomial P .A; t / of the Frobenius endomorphism  of A is a monic polynomial of
                                                              1
degree 2g in ZŒt , and its roots all have absolute value q 2 (9,16). Also, that P .A; t/ is
the characteristic polynomial of  acting on V` A. Now consider an abelian variety A over
a number field K, and assume A has good reduction at v. Let A.v/ be the corresponding
abelian variety over k.v/, and define

                                   Pv .A; t/ D P .A.v/; t/:

For any prime w lying over v, the isomorphism V` .A/ ! V` .A.v// is compatible with the
map D.w/ ! Gal.k.w/=k.v//. Since the canonical generator of Gal.k.w/=k.v// acts on
V` A.v/ as  (this is obvious from the definition of ), we see that Frobw acts on V` A as
, and so Pv .A; t / is the characteristic polynomial of Frobw acting on V` A. If w 0 also lies
over v, then Frobw 0 is conjugate to Frobw , and so it has the same characteristic polynomial.
T HEOREM 3.7. Let A and B be abelian varieties of dimension g over a number field K.
Let S be a finite set of primes of K containing all primes at which A or B has bad reduction,
and let ` be a prime different from the residue characteristics of the primes in S. Then there
exists a finite set of primes T D T .S; `; g/ of K, depending only on S , `, and g and disjoint
from S [ fv j vj`g, such that

                    Pv .A; t / D Pv .B; t/ all v 2 T H) A; B isogenous.

P ROOF. Recall: (a) A, B have good reduction at v 2 S ) V` A, V` B are unramified
at v 2 S (provided v - `) (see 3.5); (b) the action of     Ddf Gal.K al =K/ on V` A is
semisimple (see 2.5; remember we are assuming Finiteness I); (c) A and B are isogenous if
V` A and V` B are isomorphic as -modules (this is the Tate conjecture 2.6). Therefore, the
theorem is a consequence of the following result concerning `-adic representations (take
V D V` A and W D V` B).                                                                  2


L EMMA 3.8. Let .V; / and .W;  / be semisimple representations of Gal.K al =K/ on Q` -
vector spaces of dimension d . Assume that there is a finite set S of primes of K such that
 and  are unramified outside S [ fv j vj`g. Then there is a finite set T D T .S; `; d / of
primes K, depending only on S, `, and d and from disjoint from S [ fv j vj`g, such that

                   Pv .A; t / D Pv .B; t/ all v 2 T H) .V; /  .W; /:
3. FINITENESS I IMPLIES FINITENESS II.                                                 143


P ROOF. According to Theorem 3.1, there are only finitely many subfields of K al containing
                     2
K, of degree  `2d over K, and unramfied outside S [ fv j vj`g. Let L be their composite
— it is finite and Galois over K and unramified outside S [ fv j vj`g. According to the
Chebotarev Density Theorem (3.3), each conjugacy class in Gal.L=K/ is the Frobenius
class .v; L=K/ of some prime v of K not in S [ fv j vj`g. We shall prove the lemma with
T any finite set of such v’s for which
                                            [
                                 Gal.L=K/ D     .v; L=K/:
                                             v2T

Let M0 be a full lattice in V , i.e., the Z` -module generated by a Q` -basis for V . Then
AutZ` .M0 / is an open subgroup of AutQ` .V /, and so M0 is stabilized by an open sub-
group of Gal.K al =K/. As Gal.K al =K/ is compact, this shows that the lattices M0 ,
   2 Gal.K al =K/, form a finite set. Their sum is therefore a lattice M stable under
Gal.K al =K/. Similarly, W has a full lattice N stable under Gal.K al =K/. By assump-
tion, there exists a field ˝  K al , Galois over K and unramified outside the primes in
S [ fv j vj`g, such that both  and  factor through Gal.˝=K/. Because T is disjoint from
S [ fv j vj`g, for each prime w of ˝ dividing a prime v of T , we have a Frobenius element
Frobw 2 Gal.˝=K/. We are given an action of Gal.˝=K/ on M and N , and hence on
M  N . Let R be the Z` -submodule of End.M /  End.N / generated by the endomor-
phisms given by elements of Gal.˝=K/. Then R is a ring acting on each of M and N , we
have a homomorphism Gal.˝=K/ ! R , and Gal.˝=K/ acts on M and N and through
this homomorphism and the action of R on M and N . Note that, by assumption, for any
wjv 2 T , Frobw has the same characteristic polynomial whether we regard it as acting on
M or on N ; therefore it has the same trace,

                             Tr.Frobw jM / D Tr.Frobw jN /:

If we can show that the endomorphisms of M  N given by the Frobw , wjv 2 T , generate
R as a Z` -module, then (by linearity) we have that

                            Tr.rjM / D Tr.rjN /; all r 2 R:

Then the next lemma (applied to R ˝ Q` / will imply that V and W are isomorphic as
R-modules, and hence as Gal.˝=K/-modules.
L EMMA 3.9. Let k be a field of characteristic zero, and let R be a k-algebra of finite
dimension over k. Two semisimple R-modules of finite-dimension over k are isomorphic
if they have the same trace.

P ROOF. This is a standard result — see Bourbaki, Algèbre Chap 8, 12, no. 1, Prop. 3.   2


   It remains to show that the endomorphisms of M  N given by the Frobw , wjv 2 T ,
generate R (as a Z` -module). By Nakayama’s lemma, it suffices to show that R=`R is
generated by these Frobenius elements. Clearly R is a free Z` -module of rank  2d 2 , and
so
                                                            2
                            #.R=`R/ < #.R=`R/  `2d :
Therefore the homomorphism Gal.˝=K/ ! .R=`R/ factors through Gal.K 0 =K/ for
                                     2
some K 0  ˝ with ŒK 0 W K  `2d . But such a K 0 is contained in L, and by assumption
therefore, Gal.K 0 =K/ is equal to fFrobw j wjv 2 T g.                                2
144                                            CHAPTER IV. FINITENESS THEOREMS


T HEOREM 3.10. Finiteness I ) Finiteness II.

P ROOF. Recall the statement of Finiteness II: given a number field K, an integer g, and
a finite set of primes S of K, there are only finitely many isomorphism classes of abelian
varieties of K of dimension g having good reduction outside S . Since we are assuming
Finiteness I, which states that each isogeny class of abelian varieties over K contains only
finitely many isomorphism classes, we can can replace “isomorphism” with “isogeny” in
the statement to be proved. Fix a prime ` different from the residue characteristics of the
primes in S, and choose T D T .S; `; g/ as in the statement of Theorem 3.7. That theorem
then says that the isogeny class of an abelian variety A over K of dimension g and with
good reduction outside S is determined by the finite set of polynomials:

                                    fPv .A; t/ j v 2 T g:

But for each v there are only finitely many possible Pv .A; t/’s (they are polynomials of
degree 2g with integer coefficients which the Riemann hypothesis shows to be bounded),
and so there are only finitely many isogeny classes of A’s.                            2



4      Finiteness II implies the Shafarevich Conjecture.
Recall the two statements:
    Finiteness II For any number field K, integer g, and finite set S of primes of K, there
are only finitely many isomorphism classes of abelian varieties over K of dimension g
having good reduction at all primes not in S .
    Shafarevich Conjecture For any number field K, integer g  2, and finite set of primes
S of K, there are only finitely many isomorphism classes of complete nonsingular curves
over K of dimension g having good reduction outside S .
    Recall from (15.1) that, for an abelian variety A over a field k, there are only finitely
many isomorphism classes of principally polarized abelian varieties .B; / over k with
B  A. Therefore, in the statement of Finiteness II, we can replace “abelian variety” with
“principally polarized abelian variety”.
    Recall that associated with any complete smooth curve C over a field k, there is an
abelian variety J.C / of dimension g D genus.C /. In fact, J.C / has a canonical principal
polarization .C /. (We noted in (18.5) that, when k D C, there is a canonical Riemann
form; for a general k, see III 6.)

P ROPOSITION 4.1. Let C be a curve over a number field K. If C has good reduction at a
prime v of K, then so also does Jac.C /.

T HEOREM 4.2 (R ATIONAL VERSION OF T ORELLI ’ S THEOREM ). Let C be a complete non-
singular curve of genus  2 over a perfect field k. The isomorphism class of C is uniquely
determined by that of the principally polarized abelian variety .J.C /; .C //.

      On combining these two results we obtain the following theorem.

T HEOREM 4.3. Finiteness II implies the Shafarevich conjecture.
5. SHAFAREVICH’S CONJECTURE IMPLIES MORDELL’S CONJECTURE.                                  145


P ROOF. Let K be an algebraic number field, and let S be a finite set of primes in K.
From (4.1) and (4.2) we know that the map C 7! .J.C /; .C // defines an injection from
the set of isomorphism classes of complete nonsingular curves of genus  2 to the set of
isomorphism classes of principally polarized abelian varieties over K with good reduction
outside S . Thus Shafarevich’s conjecture follows from the modified version of Finiteness
II.                                                                                     2


P ROOF ( OF 4.1) We are given a complete nonsingular curve C over K that reduces to a
complete nonsingular curve C.v/ over the residue field k.v/. Therefore we have Jacobian
varieties J.C / over K and J.C.v// over k.v/, and the problem is to show that J.C / reduces
to J.C.v// (and therefore has good reduction). It is possible to do this using only varieties,
but it is much more natural to use schemes. Let R be the local ring corresponding to the
prime ideal pv in OK . To say that C has good reduction to C.v/ means that there is a
proper smooth scheme C over Spec R whose general and special fibres are C and C.v/
respectively. The construction of the Jacobian variety sketched in (17) works over R (see
JV, 8), and gives us an abelian scheme J .C/ over Spec R whose general and special fibres
are J.C / and J.C.v//, which is what we are looking for.                                    2


P ROOF ( OF 4.2) The original Torelli theorem applied only over an algebraically closed
field and had no restriction on the genus (of course, Torelli’s original paper (1914-15) only
applied over C). The proof over an algebraically closed field proceeds by a combinatorial
study of the subvarieties of C .r/ , and is unilluminating (at least to me, even my own exposi-
tion in JV, 13). Now consider two curves C and C 0 over a perfect field k, and suppose that
there is an isomorphism ˇW J.C / ! J.C 0 / (over k) sending the polarization .C / to .C 0 /.
Then the original Torelli theorem implies that there is an isomorphism W C ! C 0 over k al .
In fact, it is possible to specify uniquely (in terms of ˇ). For any  2 Gal.k al =k/, the
map of curves associated with ˇ is  . But ˇ D ˇ (this is what it means to be defined
over k), and so  D , which implies that it too is defined over k (III 12, for the details).2

E XERCISE 4.4. Does (4.2) hold if we drop the condition that g  2‹ Hints: A curve of
genus 0 over a field k, having no point in k, is described by a homogeneous quadratic
equation in three variables, i.e., by a quadratic form in three variables; now apply results on
quadratic forms (e.g., CFT, Chapter VIII). If C is a curve of genus 1 without a point, then
Jac.C / is an elliptic curve (with a point).

R EMARK 4.5. Torelli’s theorem (4.2) obviously holds for curves C of genus < 2 over k
for which C.k/ ¤ ; — a curve of genus zero with C.k/ ¤ ; is isomorphic to P1 ; a curve
of genus one with C.k/ ¤ ; is its Jacobian variety.


5    Shafarevich’s Conjecture implies Mordell’s Conjecture.
In this section, we write .f / for the divisor div.f / of a rational function on a curve. A
p-adic prime of a number field is a prime dividing p; a dyadic prime is a prime dividing 2.
    The proof that Shafarevich’s conjecture implies Mordell’s conjecture is based on the
following construction (of Kodaira and Parshin).
T HEOREM 5.1. Let K be a number field and let S be a finite set of primes of K containing
the dyadic primes. For any complete nonsingular curve C of genus g  1 over K having
146                                              CHAPTER IV. FINITENESS THEOREMS


good reduction outside S , there exists a finite extension L of K with the following property:
for each point P 2 C.K/ there exists a curve CP over L and a finite map 'P W CP ! C=L
(defined over L) such that:
   (i) CP has good reduction outside fw j wjv 2 Sg;
  (ii) the genus of CP is bounded;
 (iii) 'P is ramified exactly at P.

      We shall also need the following classical result.

T HEOREM 5.2 ( DE F RANCHIS ). Let C 0 and C be curves over a field k. If C has genus
 2, then there are only finitely many nonconstant maps C 0 ! C .

    Using (5.1) and (5.2), we show that Shafarevich’s conjecture implies Mordell’s conjec-
ture. For each P 2 C.K/, choose a pair .CP ; 'P / as in (5.1). Because of Shafarevich’s
conjecture, the CP fall into only finitely many distinct isomorphism classes. Let X be
a curve over L. If X  CP for some P 2 C.K/, then we have a nonconstant map
            'P
X  CP ! C=L ramified exactly over P , and if X  CQ , then we have nonconstant map
X ! C=L ramified exactly over Q — if P ¤ Q, then the maps differ. Thus, de Franchis’s
Theorem shows that map sending .CP ; 'P / to the isomorphism class of CP is finite-to-one,
and it follows that C.K/ is finite.                                         p
     Before proving 5.1, we make some general remarks. When is QŒ f  unramified at
p ¤ 2? Exactly when ordp .f / is odd. This is a general phenomenon: if K is the field       p of
fractions of a discrete valuation ring R and the residue characteristic is ¤ 2, then KŒ f 
is ramified if and only if ord.f / is odd. (After a change of variables, Y 2 f will be an
Eisenstein polynomial if ord.f / is odd, and will be of the form Y 2 u with u a unit if
ord.f / is even. In the second case, the discriminant is a unit.)
     Consider a nonsingular curve C over an algebraically closed field k of characteristic
¤ 2, and let f be a nonzero rational function on C . Then there is a unique nonsingular
curve C 0 over k and finite map Cp  0 ! C such that the corresponding map k.C / ,! k.C 0 /

is the inclusion k.C / ,! k.C /Œ f . Moreover, when we write .f / D 2D C D 0 with
D 0 having as few terms as possible, the remark in the preceding paragraph shows that ' is
ramified exactly at the points of support of D 0 . (If C is affine, corresponding to the ring R,
        0
                                                                           p
then C is affine, corresponding to the integral closure of R in k.C /Œ f .) For example,
consider the case when C is the affine line A1 , and let f .X/ 2 kŒX. Write

                        f .X / D f1 .X/  g.X/2 ;    f1 .X/ square-free.
             df
Then k.C 0 / D k.X /Œ f  D k.X /Œ f1 , and the curve
                     p            p


                                      C0 W   Y 2 D f1 .X/

is nonsingular because f1 .X / does not have repeated roots. The map C 0 ! C , .x; y/ 7! x,
is finite, and is ramified exactly over the roots of f1 .X/.
     When in this last example, we replace the algebraically closed field k with Q, one
additional complication occurs: f might be constant, say f D r, r 2 Q. Then C 0 !
Specm Q is the composite

                                CQŒpr ! C ! Specm CQŒpr
5. SHAFAREVICH’S CONJECTURE IMPLIES MORDELL’S CONJECTURE.                               147


— here C 0 is not geometrically connected. This doesn’t happen if ordP .f / is odd for some
point P of C .
    Next fix a pair of distinct points P1 , P2 2 A1 .Q/, and let f 2 Q.X/ be such that
.f / D P1 P2 . Construct the C 0 corresponding to f . Where does C 0 have good reduction?
Note we can replace f with cf for any c 2 Q without changing its divisor. If we want C 0
to have good reduction on as large a set as possible, we choose

                                 f D .X     P1 /=.X    P2 /

rather than, say, (*)
                                 f D p.X     P1 /=.X   P2 /:
The curve
                                 Y 2 D .X    P1 /=.X    P2 /
has good reduction at any prime where P1 and P2 remain distinct (except perhaps 2). After
these remarks, the next result should not seem too surprising.

L EMMA 5.3. Let C be a complete nonsingular curve over a number field K, and consider
a principal divisor of the form

                            P1    P2 C 2D;     P1 ; P2 2 C.K/:

Choose an f 2 K.C / such that .f / D P1 P2 C 2D, and let 'W C 0 ! C be      p the finite
covering of nonsingular curves corresponding to the inclusion K.C / ,! K.C /Œ f . With
a suitable choice of f , the following hold:
  (a) The map ' is ramified exactly at P1 and P2 .
  (b) Let S be a finite set of primes of K containing those v at which C has bad reduction,
      those v at which P1 and P2 become equal, and all primes dividing 2. If the ring of
      S-integers is a principal then C 0 has good reduction at all the primes in S.

P ROOF. (a) We have already seen this—it is really a geometric statement. (b) (Sketch.) By
assumption C extends to smooth curve C over Spec.R/, where R is the ring of S -integers.
                            df
The Zariski closure of D 0 D P1 P2 C 2D in C is a divisor on C without any “vertical
components”, i.e., without any components containing a whole fibre C.v/ of C ! Spec.R/.
We can regard f as a rational function on C and consider its divisor as well. Unfortunately,
as in the above example (*), it may have vertical components. In order to remove them we
have to replace f with a multiple by an element c 2 K having exactly the correct value
ordp .c/ for every prime ideal p in R. To be sure that such an element exists, we have to
assume that R is principal.                                                               2


R EMARK 5.4. (Variant of the lemma.) Recall that the Hilbert class field K HCF of K is a
finite unramified extension in which every ideal in K becomes principal. Even if the ring
of S -integers is not principal, there will exist an f as in the theorem in K HCF .C /.

P ROPOSITION 5.5. Let A be an abelian variety over a number field K with good reduction
outside a set of primes S. Then there is a finite extension L of K such that A.K/  2A.L/.
148                                                  CHAPTER IV. FINITENESS THEOREMS


P ROOF. The Mordell-Weil Theorem implies that A.K/=2A.K/ is finite, and we can choose
L to be any field containing the coordinates of a set of representatives for A.K/=2A.K/.
[In fact, the proposition is more elementary than the Mordell-Weil Theorem — it is proved
in the course of proving the Weak Mordell-Weil Theorem.                                 2


P ROOF ( OF 5.1) If C.K/ is empty, there is nothing to prove. Otherwise, we choose a
rational point and use it to embed C into its Jacobian. The map 2J W J ! J is étale of
degree 22g (see 7.2). When we restrict the map to the inverse image of C , we get a covering
'W C 0 ! C that is étale of degree 22g . I claim that C 0 has good reduction outside S , and
that each point of ' 1 .P / has coordinates in a field L that is unramfied over K outside
S. To see this, we need to use that multiplication by 2 is an étale map J ! J of abelian
schemes over Spec RS (RS is the ring of S -integers in K). The inclusion C ,! J extends
to an inclusion C ,! J of schemes smooth and proper over Spec RS , and fibre product of
                                              df
this with 2W J ! J gives an étale map C 0 D C J J ! C. Therefore C 0 ! Spec RS is
smooth (being the composite of an étale and a smooth morphism), which means that C 0 has
good reduction outside S . The point P defines an RS -valued point Spec.RS / ! C, and
the pull-back of C 0 ! C by this is a scheme finite and étale over Spec.RS / whose generic
fibre is ' 1 .P / — this proves the second part of the claim. For any Q 2 ' 1 .P /, ŒK.Q/ W
K  22g . Therefore, according to Theorem 3.1, there will be a finite field extension L1 of
K such that all the points of ' 1 .P / are rational over L1 for all P 2 C.K/. Now choose
two distinct points P1 and P2 lying over P , and consider the divisor P1 P2 . According
to (5.5), for some finite extension L2 of L1 , every element of J.L1 / lies in 2J.L2 /: In
particular, there is a divisor D on C=L2 such that 2D  P1 P2 . Now replace L2 with its
Hilbert class field L3 . Finally choose an appropriate f such that .f / D P1 P2 2D,
and extract a square root, as in (5.3,5.4). We obtain a map 'P
                                                      '
                                    CP ! C=L3 ! C=L3

over L3 of degree 2  22g that is ramified exactly over P . Now the Hurwitz genus formula
                                                            X
                  2 2g.CP / D .2 2g.C //  deg ' C              .eQ 1/
                                                                  Q7!P

shows that g.CP / is bounded independently of P . The field L3 is independent of P , and
(by construction) CP has good reduction outside the primes lying over S. Thus the proof
of Theorem 5.1 is complete.                                                            2


P ROOF ( OF 5.2) The proof uses some algebraic geometry of surfaces (Hartshorne, Chapter
                                                                                df
V). Consider a nonconstant map 'W C 0 ! C of curves. Its graph '  C 0  C D X is a
curve isomorphic to C 0 , and is therefore of genus g 0 D genus C 0 . Note that
                0
         '  .fP g  C / D 1;       '  .C
                                             0
                                                  fP g/ D #' 1 .P / D d;           d D deg.'/:

The canonical class of X is

                    KX  .2g       2/.C 0  fP g/ C .2g 0         2/.fP 0 g  C /

and so
                                '  KX D .2g        2/d C .2g 0      2/:
6. THE FALTINGS HEIGHT.                                                                   149


But the adjunction formula (Hartshorne V.1.5) states that
                                                  0         2
                                    '  KX D 2g       2     ':

We deduce that
                                       2
                                       ' D      .2g   2/d
which is negative, because of our assumption on g D g.C /. Note that d is bounded: the
Hurwitz formula says that

                              2g   2 D d.2g 0     2/ C .positive/:

Thus, there is an integer N (independent of ') such that
                                                2
                                        N      ' < 0:

For each polynomial P there exists a Hilbert scheme, HilbP ; classifying the curves on X
with Hilbert polynomial P . We know that HilbP is a finite union of varieties Vi (when the
ambient space is Pn , it is even connected), and that if 2 Vi , then dim Vi D dim H 0 . ; N /
(by deformation theory) where N is the normal bundle. In our case, N D 0 since
  2
  ' < 0. Thus each Vi is a point. We deduce that

                                f ' j ' 2 Homnoncnst .C; C 0 /g

is finite, and since a map is determined by its graph, this proves the theorem. Alterna-
tive approach: Use differential geometry. The condition g.C / > 1 implies that C.C/ is
hyperbolic.                                                                           2



6     The Faltings Height.
To any abelian variety A over a number field K, Faltings attaches a canonical height
H.A/ 2 R.

The Faltings height of an elliptic curve over Q
Consider first an elliptic curve E over C. We want to attach a number H.E/ to E which
is a measure of its “size”. The most natural first attempt would be to write E  C=,
and define H.E/ to be the reciprocal of the area of a fundamental domain for , i.e., if
 D Z!1 C Z!2 , then
                                   H.E/ D j!1 ^ !2 j 1 :
Unfortunately this doesn’t make sense, because we can scale the isomorphism to make the
area of the fundamental domain any positive real number we choose. In order to get a
height, we need additional data.

P ROPOSITION 6.1. Let E be an elliptic curve over C. Then each of the following choices
determines the remainder:
    (a) an isomorphism C= ! E.C/;
    (b) the choice of a basis for .E; ˝ 1 /, i.e., the choice of a nonzero holomorphic differ-
        ential on E;
150                                               CHAPTER IV. FINITENESS THEOREMS


  (c) the choice of an equation
                                  Y 2 D 4X 3      g2 X   g3      (*)
      for E.

P ROOF. (a)!(c). There are associated with a lattice , a Weierstrass function }.z/ and
numbers g2 ./, g3 ./ for which there is an isomorphism
                 E.C/ D C= ! E 0  P2 ;             z 7! .}.z/ W } 0 .z/ W 1/
where E 0 is the projective curve given by the equation (*).
    (c) ! (b). Take ! D dX   Y .
    (b) ! (a). From a differential ! on E and an isomorphism ˛W C= ! E.C/ we obtain
a differential ˛  .!/ on C invariant under translation by elements of . For example, if ˛ is
                                                    d }.z/
the map given by } and ! D dX              
                                 Y , then ˛ .!/ D } 0 .z/ D dz. Thus we should choose the ˛
                                                                           RP
so that ˛  .!/ D dz. This we can do as follows: consider the map P 7! 0 !W E.C/ ! C.
This is not well-defined because the integral depends on the choice of the path. However, if
 1 and 2 are generators for H1 .E; Z/, then (up to homotopy), two paths from 0 to P will
differ by a loop m1 1 C m2 2 , and because ! is holomorphic, the integral depends only on
the homotopy class of theR path. Therefore, we obtain a well-defined map E.C/ ! C=,
 D Z!1 C Z!2 , !i D i !, which is an isomorphism.                                          2


  Now, given a pair .E; !/ over C, we can define
             i                i                i
               Z                Z                Z                   Z
H.E; !/ 1 D          !^!N D         dz^d zN D      d.xCiy/^d.x iy/ D     dx^dy
            2 E.C/            2 D              2 D                     D

where D is a fundamental domain for . Thus H.E; !/ 1 is the area of D.
   When the elliptic curve is given over Q (rather than C), then we choose an equation
                          Y 2 D 4X 3      g2 X    g3 ;   g2 ; g3 2 Q;
and take the differential ! to be dX=Y . When we change the choice of the equation, ! is
only multiplied by a nonzero rational number, and so
                                             df
                                     H.E/ D H.E; !/
is a well-defined element of RC =QC , but we can do better: we know that E has a global
minimal model i.e., an equation
       Y 2 C a1 X Y C a3 Y D X 3 C a2 X 2 C a4 X C a6 ;         ai 2 Z;      minimal.
The Weierstrass .DNéron) differential,
                                               dX
                                   !D                    :
                                          2Y C a1 X C a3
is well-defined up to a multiplication by a unit in Z, i.e., up to sign. Now H.E/ D H.E; !/
is uniquely determined.
    When we consider an elliptic curve over a number field K two complications arise.
Firstly, K may have several infinite primes, and so we may have to take the product over
their separate contributions. Secondly, and more importantly, OK may not be a principal
ideal domain, and so there may not be a global minimal equation. Before describing how
to get around this last problem, it is useful to consider a more general construction.
6. THE FALTINGS HEIGHT.                                                                    151


The height of a normed module
A norm on a vector space M over R or C is a mapping k  kW M ! R>0 such that
            kx C yk  kxk C kyk;         kaxk D jajkxk;            x; y 2 M;   a scalar.
Here j  j is the usual absolute value.
     Now let K be a number field, and let R be the ring of integers in K. Recall that a
fractional ideal in K is a projective R-module of rank 1; conversely, if M is a projective
R-module of rank 1, then M ˝R K  K, and the choice of an isomorphism identifies M
with a fractional ideal in K. Let M be such an R-module. Suppose we are given a norm
k  kv on M ˝R Kv for each vj1. We define the height of M (better, of .M; .k  kv /vj1 /)
to be
                                                                    
                 .M W Rm/                                              1 v real
   H.M / D Q               "v ;  m any nonzero element of M; "v D
                  vj1 kmkv
                                                                       2 v complex.
L EMMA 6.2. The definition is independent of the choice of m.

P ROOF. Recall that, for a finite prime v corresponding to a prime ideal p, the normalized
absolute value is defined by,
                                           ordv .a/
                          jajv D .R W p/              ;       ordv W K  Z;
and that for any infinite prime v,
                                         jajv D jaj"v :
Moreover, for the normalized absolute values, the product formula holds:
                                      Y
                                         jajv D 1:
The Chinese remainder theorem shows that
                                M=Rm  ˚v finite Mv =Rv m
where Rv is the completion of R at v and Mv D Rv ˝R M . Now Mv is a projective module
of rank 1 over Rv , and hence it is free of rank 1 (because Rv is principal), say Mv D Rv mv .
Therefore                                                      ˇm ˇ
                                                               ˇ vˇ
                         .Mv W Rv m/ D .Rv mv W Rv m/ D ˇ           ˇ ;
                                                                m v
where by mv =m we mean the unique element a of Kv such that am D mv . Hence we find
that
                                                       1
                          H.M / D Q              ˇ ˇ Q                :                   (6)
                                                 ˇmˇ               "v
                                         v finite mv
                                                 ˇ   ˇ   vj1 kmk  v
                                                          v
It is obvious that the expression on the right is unchanged when m is replaced with am.     2


L EMMA 6.3. In the expression (6) for H.M /, m can be taken to be any element of M ˝R
K. When we define,
                                           1
                                h.M / D         log H.M /;
                                        ŒK W Q
then, for any finite extension L of K,
                                     h.RL ˝R M / D h.M /:

P ROOF. Exercise in algebraic number theory.                                                2
152                                              CHAPTER IV. FINITENESS THEOREMS


The Faltings height of an abelian variety
P ROPOSITION 6.4. Let V be a smooth algebraic variety of dimension g over a field k.
  (a) The sheaf of differentials ˝V1 =k on V is a locally free sheaf of OV -modules of rank
      g.
  (b) If V is a group variety, then ˝ 1 is free.

P ROOF. See T. Springer, Linear Algebraic Groups, Birkhäuser, 1981, 3.2, 3.3.                 2


C OROLLARY 6.5. Let V be a smooth algebraic variety of dimension g over a field k. Then
˝ g Ddf g ˝ 1 is a locally free sheaf of rank 1, and it is free if V is a group variety.

P ROOF. Immediate from (6.4).                                                                  2


    Let M be a coherent sheaf on a variety V . For any point v 2 V we obtain a vector space
M.v/ over the residue field k.v/. For example, if V is affine, say V D Specm.R/, then M
corresponds to the R-module M D .V; M/, and if v $ m, then M.v/ D M=mM . Note
that, for any open subset U of V containing v, there is a canonical map .U; M/ ! M.v/.

P ROPOSITION 6.6. Let V be a complete geometrically connected variety over a field k,
and let M be a free sheaf of finite rank on V . For any v 2 V .k/, the map .V; M/ !
M.v/ is an isomorphism.

P ROOF. For M D OV , .V; M/ D k (the only functions regular on the whole of a
complete variety are the constant functions), and the map is the identity map k ! k. By
assumption M  .OV /n for some n, and so the statement is obvious.                    2


P ROPOSITION 6.7. Let A be an abelian variety of dimension g over a field k. The canoni-
cal maps
                    .A; ˝ 1 / ! ˝ 1 .0/;        .A; ˝ g / ! ˝ g .0/
are isomorphisms.

P ROOF. By 0 we mean the zero element of A. For the proof, combine the last two results.2

     Now let A be an abelian variety over a number field K, and let R be the ring of integers
in K. Recall from I, 17, that there is a canonical extension of A to a smooth group scheme
                                                  g
A over Spec R (the Néron model). The sheaf ˝A=R       of (relative) differential g-forms on A
is a locally free sheaf of OA -modules of rank 1 (it becomes free of rank 1 when restricted
to each fibre, but is not free on the whole of A). There is a section sW Spec R ! A whose
                                                              g
image in each fibre is the zero element. Define M D s  ˝A=R      . It is a locally free sheaf of
rank 1 on Spec R, and it can therefore be regarded as a projective R-module of rank 1. We
have
                                            g                   g
                            M ˝R K D ˝A=K      .0/ D .A; ˝A=K       /
                                       g
—the first equality simply says that ˝A=R   restricted to the zero section of A and then to the
                            g
generic fibre, is equal to ˝A=R restricted to the generic fibre, and then to the zero section;
the second equality is (6.7).
7. THE MODULAR HEIGHT.                                                                  153


   Let v be an infinite prime of K. We have to define a norm on M ˝K Kv . But M ˝K
                 g
Kv D .AKv ; ˝A     K =Kv
                         /, and we can set
                     v

                                    g Z                  12
                                     i
                            k!kv D                  ! ^ !N      :
                                     2     A.Kval /

Note that Kval D C. Now .M; .k  kv // is a normed R-module, and we define the Faltings
height of A,
                                   H.A/ D H.M /:
    We can make this more explicit by using the expression (26.2.1) for H.M /. Choose
a holomorphic differential g-form ! on A=K—this will be our m. It is well-defined up
to multiplication by an element of K  . For a finite prime v, we have a Néron differential
g-form !v for A=Kv (well-defined up to multiplication by a unit in Rv ), and we have
                                                1
                 H.A/ D          ˇ   ˇ          g R              "v =2
                            Q    ˇ!ˇ Q          i
                              v-1 !v
                                 ˇ   ˇ  vj1    2         al
                                                       A.Kv / ! ^ N
                                                                  !

For any infinite prime v, choose an isomorphism

                                    ˛W Cg = ! A.Kval /

such that ˛  .!/ D dz1 ^ dz2 ^ : : : ^ dzg ; then the contribution of the prime v is
                                                                    "v
                         .volume of a fundamental domain for / 2 :

This is all very explicit when A is an elliptic curve. In this case, !v is the differential
corresponding to the Weierstrass minimal equation (see above, and Silverman 1986, VII.1).
There is an algorithm for finding the Faltings height of an elliptic curve, which has surely
been implemented for curves over Q (put in the coefficients; out comes the height).
    Define
                                            1
                                h.A/ D           log H.A/:
                                         ŒK W Q
If L is a finite extension of K, it is not necessarily true that h.AL / D h.A/ because the
Néron minimal model may change (Weierstrass minimal equation in the case of elliptic
curves). However, if A has semistable reduction everywhere, then h.A/ is invariant under
finite field extensions. We define the stable Faltings height of A,

                                      hF .A/ D h.AL /

where L is any finite field extension of K such that AL has semistable reduction at all
primes of L (see I 17.3).


7    The Modular Height.
Heights on projective space
(Serre, 1989, 2). Let K be a number field, and let P D .x0 W : : : W xn / 2 Pn .K/. The
height of P is defined to be            Y
                               H.P / D       max jxi jv :
                                                 0i n
                                             v
154                                              CHAPTER IV. FINITENESS THEOREMS


Define
                                              1
                                 h.P / D           log H.P /:
                                           ŒK W Q
         Serre puts the factor ŒK W Q into H.P /.
P ROPOSITION 7.1. For any number C , there are only finitely many points P of Pn .K/
with H.P /  C .

    Note that an embedding ˛W V ,! Pn of an algebraic variety into Pn defines on it a
height function, H.P / D H.˛.P //:
P ROPOSITION 7.2. Let ˛1 and ˛2 be two embedding of V into Pn such that ˛1 1 .hyperplane/ 
˛2 1 .hyperplane). Then the height functions defined by ˛1 and ˛2 on V differ by a bounded
amount.

   In other words, given a variety V and a very ample divisor on V , we get a height
function on V .K/, well defined up to a bounded function.

The Siegel modular variety
For any field L, let Mg;d .L/ be the set of isomorphism classes of pairs .A; / with A an
abelian variety over L of dimension g and  a polarization of A of degree d .
T HEOREM 7.3. There exists a unique algebraic variety Mg;d over C and a bijection

                                  j W Mg;d .C/ ! Mg;d .C/

such that:
  (a) for every point P 2 Mg;d , there is an open neighbourhood U of P and a family A
      of polarized abelian varieties over U such that the fibre AQ represents j 1 .Q/ for
      all Q 2 Mg;d ;
  (b) for any variety T over C, and family A of polarized abelian varieties over T of
      dimension g and degree d , the map T ! Mg;d , t 7! j.At /, is regular (i.e., is a
      morphism of algebraic varieties).

P ROOF. Uniqueness: Let .M 0 ; j 0 / be a second pair, and consider the map j 0 ıj W Mg;d .C/ !
M 0 .C/. To prove that this is regular, it suffices to prove that it is regular in a neighbourhood
of each point P of Mg;d . But given P , we can find a neighbourhood U of P as in (a),
and condition (b) for M 0 implies that .j 0 ı j /jU is regular. Similarly, its inverse is regular.
Existence: This is difficult. Siegel constructed Mg;d as a complex manifold, and Satake
and others showed about 1958 that it was an algebraic variety. See E. Freitag, Siegelsche
Modulfunktionen, Springer, 1983.                                                                 2


      The variety in the theorem is called the Siegel modular variety.
E XAMPLE 7.4. The j -invariant defines a bijection

           felliptic curves over Cg= D M1;1 .C/ ! M1;1 .C/;             M1;1 D A1 :

See, for example, Milne 2006, V 2.2.
7. THE MODULAR HEIGHT.                                                                    155


   Note that the automorphisms of C act on Mg;d .C/.
   Let V be a variety over C, and suppose that there is given a model V0 of V over Q (AG,
Chapter 16). Then the automorphisms of C act on V0 .C/ D V .C/.

T HEOREM 7.5. There exists a unique model of Mg;d over Q such the bijection j W Mg;d .C/ !
Mg;d .C/ commutes with the two actions of Aut.C/ noted above.

    Write Mg;d again for this model. For each field L  Q, there is a well-defined map

                                       j W Mg;d .L/ ! Mg;d .L/

that is functorial in L and is an isomorphism whenever L is algebraically closed.

P ROOF. This is not difficult4 , given (7.3).                                               2


E XAMPLE 7.6. The model of M1;1 over Q is just A1 again. The fact that j commutes with
the actions of Aut.C/ simply means that, for any automorphism  of C and elliptic curve
E over C, j.E/ D j.E/—if E has equation

                                          Y 2 D X 3 C aX C b

then E has equation
                                        Y 2 D X 3 C aX C b;
and so this is obvious.
    Note that j W M1;1 .L/ ! A1 .L/ D L will not in general be a bijection unless L is
algebraically closed. For example, if c 2 L, then the curve

                                  Ec W      Y 2 D X 3 C ac 2 X C bc 3

has the same j -invariant as

                                      EW       Y 2 D X 3 C aX C b

but it is not isomorphic to E over L unless c is a square in L.

R EMARK 7.7. Let K be a number field, and consider the diagram:
                                                       j
                                   Mg;d .K/                !    Mg;d .K/
                                     ?                             ?
                                     ?                             ?injection
                                     y                             y
                                                       j
                                  Mg;d .K al /             ! Mg;d .K al /
                                                    bijection

     Clearly j.A; / D j.A0 ; 0 / if and only if .A; / becomes isomorphic to .A0 ; 0 / over
K al .
   4 That’s what my original notes say, but I’m not sure I believe it.
156                                             CHAPTER IV. FINITENESS THEOREMS


The modular height
The proof that Mg;d is an algebraic variety shows more, namely, that there is a canonical
ample divisor on Mg;d , and therefore a height function h on Mg;d .K/, any number field
K, well-defined up to a bounded function, and we define the modular height of a polarized
abelian variety .A; / over K by

                                  hM .A; / D h.j.A; //:

For example, consider the elliptic curve E over Q; write j.E/ D m
                                                                n with m and n relatively
prime integers. Then hM .E/ D log maxfjmj; jnjg.

T HEOREM 7.8. For every polarized abelian variety .A; / over a number field K,

                         hF .A/ D hM .A; / C O.log hM .A; //:

P ROOF. Technically, this is by far the hardest part of the proof. It involves studying the two
height functions on a compactification of the modular variety over Z (see Chai and Faltings
1990 for moduli schemes over Z).                                                              2


E XERCISE 7.9. Prove (7.8) for elliptic curves.

T HEOREM 7.10. Let K be a number field, and let g; d , C be integers. Up to isomorphism,
there are only finitely many polarized abelian varieties .A; / over K of dimension g and
degree d with semistable reduction everywhere and

                                        hM .A; /  C:

R EMARK 7.11. The semistability condition is essential, for consider an elliptic curve

                                EW      Y 2 D X 3 C aX C b

over K. For any c 2 K  , c … K 2 ,

                             Ec W      Y 2 D X 3 C ac 2 X C bc 3

has the same height as E (because it has the same j -invariant), but it is not isomorphic to
E over K.

P ROOF. (of Theorem 7.10). We know from (7.1) that

                               fP 2 Mg;d .K/ j H.P /  C g

is finite, and we noted above, that .A; / and .A0 ; 0 / define the same point in Mg;d .K/ if
and only if they become isomorphic over K al . Therefore, it suffices to prove the following
statement:
        Let .A0 ; 0 / be a polarized abelian variety over K with semistable reduction
        everywhere; then up to K-isomorphism, there are only finitely many .A; /
        over K with semistable reduction everywhere such that .A; /  .A0 ; 0 /
        over K al .
7. THE MODULAR HEIGHT.                                                                     157


    Step 1. Let S be the set of primes of K at which A0 has bad reduction, and let A be as
in the statement. Then S is also the set of primes where A has bad reduction. Proof: We
know that A and A0 become isomorphic over a finite extension L of K. Because A0 and
A have semstable stable reduction everywhere, when we pass from K to L, bad reduction
stays bad reduction and good reduction stays good reduction, and so the set of primes of K
where A has bad reduction can be read off from the similar set for L.
    Step 2. Now fix an `  3. There exists a finite extension L of K such that all the A’s in
the statement have their points of order ` rational over L.
    Proof: The extension K.A` / of K obtained by adjoining the points of order ` is an
extension of K of degree  # GL2g .F` /Z=`Z/ unramified outside S and fv j vj`g (see
3.5), and so we can apply (3.1).
    Step 3. Every .A; / as in the statement becomes isomorphic to .A0 ; 0 / over the field
L in the Step 2.
    Proof: Recall (I 14.4) that any automorphism of .A; / that acts as the identity map on
the points of order 3 is itself the identity map. We are given that there is an isomorphism
˛W .A; / ! .A0 ; 0 / over K al . Let  2 Gal.K al =L/. Then ˛ is a second isomorphism
.A; / ! .A0 ; 0 /, and  ˛ and ˛ have the same action on the points of order 3 of A. (By
definition . ˛/.P / D  .˛. 1 P /, but because  fixes L,  1 P D P and .˛P / D ˛P .)
Hence  ˛ ı ˛ 1 is an automorphism of .A0 ; 0 / fixing the points of order 3, and so it is the
identity map. Therefore  ˛ D ˛, and this means ˛ is defined over L.
    Step 4. Take the field L in step 2 to be Galois over K. Then there is a canonical bijection
between the following two sets:

                  f.A; / j .A; /  .A0 ; 0 / over Lg=.K-isomorphism/
                          H 1 .Gal.L=K/; Aut.AL ; L //:

Proof: Given .A; /, choose an isomorphism ˛W .A; / ! .A0 ; 0 / over L, and let

                            a D  ˛ ı ˛ 1 ;      2 Gal.L=K/:

Then  7! a is a crossed homomorphism Gal.L=K/ ! Aut.AL ; L /, and it is not dif-
ficult to prove that the map sending .A; / to the cohomology class of .a / is a bijection.
The group H 1 .Gal.L=K/; Aut.AL ; L // is finite because Gal.L=K/ and Aut.AL ; L //
are both finite (for the second group, see I 14.4), and this completes the proof of the Theo-
rem.                                                                                       2


C OROLLARY 7.12. Let K be a number field, and let g, d , C be integers. Up to isomor-
phism, there are only finitely many polarized abelian varieties .A; / over K of dimension
g and degree d with semistable reduction everywhere and

                                        hF .A/  C:

P ROOF. Apply (7.8).                                                                         2


C OROLLARY 7.13. Let K be a number field, and let g; C be integers. Up to isomorphism,
there are only finitely many abelian varieties A over K of dimension g with semistable
reduction everywhere and
                                      hF .A/  C:
158                                            CHAPTER IV. FINITENESS THEOREMS


P ROOF. We need one more result, namely that hF .A/ D hF .A_ /. (This is proved by Ray-
                                                               df
naud in the Szpiro seminar.) Given an A as in the statement, B D .AA_ /4 is a principally
polarized abelian variety over K (see I 13.12) with semistable reduction everywhere, and

                                   h.B/ D 8h.A/  8C:                                      2

Therefore we can apply (7.12) (and I 15.3).




8     The Completion of the Proof of Finiteness I.
It remains to prove:
         Finiteness I: Let A be an abelian variety over a number field K. There are
      only finitely many isomorphism classes of abelian varieties B over K isoge-
      nous to A.

T HEOREM 8.1. Let A be an abelian variety over a number field K having semistable re-
duction everywhere. The set of Faltings heights of abelian varieties B over K isogenous to
A is finite.

    Before discussing the proof of (8.1), we explain how to deduce Finiteness I. First as-
sume that A has semistable reduction everywhere. Then so also does any B isogenous to
A, and so (7.13) and (8.1) show that the set of isomorphism classes of such B’s is finite.
    Now consider an arbitrary A. There will be a finite extension L of K such that A
acquires semistable reduction over L, and so Finiteness I follows from the next statement:
up to isomorphism, there are only finitely many abelian varieties B over K isogenous to a
fixed abelian variety B0 over K, and isomorphic to B0 over L. (Cf. the proof of the last
step of 7.10.)

P ROOF ( OF 8.1) Faltings’s original proof used algebraic geometry, and in particular a the-
orem of Raynaud’s on finite group schemes. In his talks in Szpiro’s seminar, Raynaud
improved Faltings’s results by making them more effective.                                2



Appendix: Review of Faltings 1983 (MR 85g:11026)
Faltings, G.,
Endlichkeitssätze für abelsche Varietäten über Zahlkörpern. [Finiteness Theorems
for Abelian Varieties over Number Fields],
Invent. Math. 73 (1983), 349-366; Erratum,ibid. (1984), 75, 381.

     The most spectacular result proved in this paper is Mordell’s famous 1922 conjecture: a
nonsingular projective curve of genus at least two over a number field has only finitely many
points with coordinates in the number field. This result is in fact obtained as a corollary
of finiteness theorems concerning abelian varieties which are themselves of at least equal
significance. We begin by stating them. Unless indicated otherwise, K will be a number
field, the absolute Galois group Gal.K=K/ of K, S a finite set of primes of K, and A an
8. THE COMPLETION OF THE PROOF OF FINITENESS I.                                           159


abelian variety over K. For a prime number l, Tl A will denote the Tate group of A (inverse
limit of the groups of l n -torsion points on A) and Vl A D Ql ˝Zl Tl A. The paper proves
the following theorems.
    T HEOREM 3. The representation of       on Vl A is semisimple.
    T HEOREM 4. The canonical map EndK .A/ ˝Z Zl ! End.Tl A/ is an isomorphism.
    T HEOREM 5. For given S and g, there are only finitely many isogeny classes of abelian
varieties over K with dimension g and good reduction outside S .
    T HEOREM 6. For given S; g, and d , there are only finitely many isomorphism classes
of polarized abelian varieties over K with dimension g, degree (of the polarization) d , and
good reduction outside S .
    Both Theorem 3 and Theorem 4 are special cases of conjectures concerning the étale co-
homology of any smooth projective variety. The first is sometimes called the Grothendieck-
Serre conjecture; the second is the Tate conjecture. Theorem 6 is usually called Shafare-
vich’s conjecture because it is suggested by an analogous conjecture of his for curves (see
below).
    In proving these theorems, the author makes use of a new notion of the height h.A/ of
an abelian variety: roughly, h.A/ is a measure of the volumes of the manifolds A.K v /, v
an Archimedean prime of K, relative to a Néron differential on A. The paper proves:
   T HEOREM 1. For given g and h, there exist only finitely many principally polarized
abelian varieties over K with dimension g, height  h, and semistable reduction every-
where.
    T HEOREM 2. Let A.K/.l/ be the l-primary component of A.K/, some prime number
l, and let G be an l-divisible subgroup of A.K/.l/ stable under . Let Gn denote the set
of elements of G killed by l n . Then, for n sufficiently large, h.A=Gn / is independent of n.
   T HEOREM (*) Let A be an abelian variety over K with semistable reduction every-
where; then there is an N such that for every isogeny A ! B of degree prime to N ,
h.A/ D h.B/.
    The proof of Theorem 4 is modelled on a proof of J. T. Tate for the case of a finite field
K [same journal 2 (1966), 134–144; MR 34#5829]. There, Tate makes use of a (trivial)
analogue of Theorem 6 for a finite field to show that a special element of End.Tl A/ lies
in the image of the map. At the same point in the proof, the author applies his Theorems 1
and 2. An argument of Yu. G. Zarkhin [Izv. Akad. Nauk SSSR Ser. Mat. 39 (1975), no. 2,
272–277; MR 51#8114] allows one to pass from the special elements to a general element.
Theorem 3 is proved simultaneously with Theorem 4.
    From Theorem 4 in the case of a finite field, it follows that the isogeny class of an
abelian variety over a finite field is determined by the characteristic polynomial of the
Frobenius element. By making an adroit application of the Chebotarev density theorem
(and Theorems 3 and 4), the author shows the following: given S and g, there exists a finite
set T of primes of K such that the isogeny class of an abelian variety over K of dimen-
sion g with good reduction outside S is determined by the characteristic polynomials of
the Frobenius elements at the v in T . (This in fact seems to give an algorithm for deciding
when two abelian varieties over a number field are isogenous.) Since the known properties
of these polynomials (work of Weil) imply there are only finitely many possibilities for each
prime, this proves Theorem 5.
    In proving Theorem 6, only abelian varieties B isogenous to a fixed abelian variety
A need be considered (because of Theorem 5), and, after K has been extended, A can be
160                                            CHAPTER IV. FINITENESS THEOREMS


assumed to have semistable reduction everywhere. The definition of the height is such that
                                   df
                          e.B=A/ D exp.2ŒK W Q.h.B/         h.A//

is a rational number whose numerator and denominator are divisible only by primes di-
viding the degree of the isogeny between A and B. Therefore (*) shows that there exists
an integer N such that e(A/B) involves only the primes dividing N . The isogenies whose
   Q are divisible only by the primes dividing N correspond to the -stable sublattices
degrees
of ljN Tl A. From what has been shown about Tl A, there exist only finitely many iso-
morphism classes of such sublattices, and this shows that the set of possible values h.B/ is
finite. Now Theorem 1 can be applied to prove Theorem 6.
     The proof of Theorem 1 is the longest and most difficult part of the paper. The basic
idea is to relate the theorem to the following elementary result: given h, there are only
finitely many points in Pn .K/ with height (in the usual sense)  h. The author’s height
defines a function on the moduli space Mg of principally polarized abelian varieties of
dimension g. If Mg is embedded in PnK by means of modular forms rational over K, then
the usual height function on Pn defines a second function on Mg . The two functions must
be compared. Both are defined by Hermitian line bundles on Mg and the main points are
to show (a) the Hermitian structure corresponding to the author’s height does not increase
too rapidly as one approaches the boundary of Mg (it has only logarithmic singularities)
and (b) by studying the line bundles on compactifications of moduli schemes over Z, one
sees that the contributions to the two heights by the finite primes differ by only a bounded
amount. This leads to a proof of Theorem 1. (P. Deligne has given a very concise, but
clear, account of this part of the paper [“Preuve des conjectures de Tate et de Shafarevitch”,
Seminaire Bourbaki, Vol. 1983/84 (Paris, 1983/84), no. 616; per revr.].)
     The proofs of Theorems 2 and (*) are less difficult: they involve calculations which
reduce the questions to formulas of M. Raynaud [Bull. Soc. Math. France 102 (1974),
241–280; MR 547488]. (To obtain a correct proof of Theorem 2, one should replace the A
of the proof in the paper by A=Gn , some n sufficiently large.)
     Torelli’s theorem says that a curve is determined by its canonically polarized Jacobian.
Thus Theorem 6 implies the (original) conjecture of Shafarevich: given S and g, there exist
only finitely many nonsingular projective curves over K of genus g and good reduction
outside S . An argument of A. N. Parshin [Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968),
1191–1219; MR 411740] shows that Shafarevich’s conjecture implies that of Mordell: to
each rational point P on the curve X one associates a covering 'P W XP ! X of X ; the
curve XP has bounded genus and good reduction outside S ; thus there are only finitely
many possible curves XP , and a classical theorem of de Franchis shows that for each XP
there are only finitely many possible 'P ; as the association P 7! .XP ; 'P / is one-to-one,
this proves that there are only finitely many P .
     Before this paper, it was known that Theorem 6 implies Theorems 3 and 4 (and Mordell’s
conjecture). One of the author’s innovations was to see that by proving a weak form of The-
orem 6 (namely Theorem 1) he could still prove Theorems 3 and 4 and then could go back
to get Theorem 6.
     Only one misprint is worth noting: the second incorrect reference in the proof of Theo-
rems 3 and 4 should be to Zarkhin’s 1975 paper [op. cit.], not his 1974 paper.
     James Milne .
Bibliography

A RTIN , M. 1986. Néron models, pp. 213–230. In Arithmetic geometry (Storrs, Conn., 1984). Springer, New
   York.

ATIYAH , M. F. AND M ACDONALD , I. G. 1969. Introduction to commutative algebra. Addison-Wesley Pub-
  lishing Co., Reading, Mass.-London-Don Mills, Ont.

BARSOTTI , I. 1953. A note on abelian varieties. Rend. Circ. Mat. Palermo (2) 2:236–257 (1954).

B IRKENHAKE , C. AND L ANGE , H. 2004. Complex abelian varieties, volume 302 of Grundlehren der Mathe-
   matischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin.

B OOTHBY, W. M. 1975. An introduction to differentiable manifolds and Riemannian geometry. Academic
   Press, New York-London.

B OREL , A. 1953. Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de
   Lie compacts. Ann. of Math. (2) 57:115–207.

B OREL , A. 1969. Introduction aux groupes arithmétiques. Publications de l’Institut de Mathématique de
   l’Université de Strasbourg, XV. Actualités Scientifiques et Industrielles, No. 1341. Hermann, Paris.

B OSCH , S., L ÜTKEBOHMERT, W., AND R AYNAUD , M. 1990. Néron models, volume 21 of Ergebnisse der
   Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Springer-Verlag,
   Berlin.

B OURBAKI , N. 1989. Commutative algebra. Chapters 1–7. Elements of Mathematics. Springer-Verlag, Berlin.

C ARTAN , H. 1963. Elementary theory of analytic functions of one or several complex variables. Éditions
   Scientifiques Hermann, Paris.

C ASSELS , J. W. S. AND F LYNN , E. V. 1996. Prolegomena to a middlebrow arithmetic of curves of genus 2,
   volume 230 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge.

C HOW, W.-L. 1954. The Jacobian variety of an algebraic curve. Amer. J. Math. 76:453–476.

C ONRAD , B. 2006. Chow’s K=k-image and K=k-trace, and the Lang-Néron theorem. Enseign. Math. (2)
   52:37–108.

F LYNN , E. V. 1990. The Jacobian and formal group of a curve of genus 2 over an arbitrary ground field. Math.
   Proc. Cambridge Philos. Soc. 107:425–441.

G ROTHENDIECK , A. 1959–62. Technique de descente et théorèmes d’existence en géométrie algébrique. I–VI.,
   pp. Exp. Nos. 190, 195, 212, 221, 232, 236. In Séminaire Bourbaki. Soc. Math. France, Paris.

G ROTHENDIECK , A. 1968. Le groupe de Brauer. III. Exemples et compléments, pp. 88–188. In Dix Exposés
   sur la Cohomologie des Schémas. North-Holland, Amsterdam. Available at www.grothendieck-circle.org.

G ROTHENDIECK , A. 1971. Revêtements étales et groupe fondamental. Springer-Verlag, Berlin. Available at
   www.grothendieck-circle.org.

G ROTHENDIECK , A. AND D IEUDONN É , J. 1971. Eléments de Géomètrie Algèbrique, I. Springer-Verlag,
   Heidelberg.


                                                     161
162                                                                                    BIBLIOGRAPHY


H ARTSHORNE , R. 1977. Algebraic geometry. Springer-Verlag, New York.

K AHN , B. 2006. Sur le groupe des classes d’un schéma arithmétique. Bull. Soc. Math. France 134:395–415.
   With an appendix by Marc Hindry.

L ANG , S. 1959. Abelian varieties. Interscience Tracts in Pure and Applied Mathematics. No. 7. Interscience
   Publishers, Inc., New York.

L ANG , S. 1983. Fundamentals of Diophantine geometry. Springer-Verlag, New York.

L ANG , S. AND N ÉRON , A. 1959. Rational points of abelian varieties over function fields. Amer. J. Math.
   81:95–118.

L ICHTENBAUM , S. 1969. Duality theorems for curves over p-adic fields. Invent. Math. 7:120–136.

M ATSUSAKA , T. 1953. Some theorems on Abelian Varieties. Nat. Sci. Rep. Ochanomizu Univ. 4:22–35.

M ATTUCK , A. AND M AYER , A. 1963. The Riemann-Roch theorem for algebraic curves. Ann. Scuola Norm.
  Sup. Pisa (3) 17:223–237.

M ILNE , J. S. 1980. Etale cohomology, volume 33 of Princeton Mathematical Series. Princeton University
   Press, Princeton, N.J.

M ILNE , J. S. 1986. Abelian varieties, pp. 103–150. In Arithmetic geometry (Storrs, Conn., 1984). Springer,
   New York.

M ILNE , J. S. 2006. Elliptic Curves. BookSurge Publishers.

M UMFORD , D. 1966. Lectures on curves on an algebraic surface. With a section by G. M. Bergman. Annals
  of Mathematics Studies, No. 59. Princeton University Press, Princeton, N.J.

M UMFORD , D. 1970. Abelian varieties. Tata Institute of Fundamental Research Studies in Mathematics, No.
  5. Published for the Tata Institute of Fundamental Research, Bombay, by Oxford University Press.

M UMFORD , D. 1999. The red book of varieties and schemes, volume 1358 of Lecture Notes in Mathematics.
  Springer-Verlag, Berlin.

M URTY, V. K. 1993. Introduction to abelian varieties, volume 3 of CRM Monograph Series. American
  Mathematical Society, Providence, RI.

N ÉRON , A. 1964. Modèles minimaux des variétés abéliennes sur les corps locaux et globaux. Inst. Hautes
    Études Sci. Publ.Math. No. 21:128.

O ORT, F. 1966. Commutative group schemes, volume 15 of Lecture Notes in Mathematics. Springer-Verlag,
   Berlin.

S ERRE , J.-P. 1959. Groupes algébriques et corps de classes. Publications de l’institut de mathématique de
   l’université de Nancago, VII. Hermann, Paris.

S ERRE , J.-P. 1962. Corps locaux. Publications de l’Institut de Mathématique de l’Université de Nancago,
   VIII. Actualités Sci. Indust., No. 1296. Hermann, Paris.

S ERRE , J.-P. 1989. Lectures on the Mordell-Weil theorem. Aspects of Mathematics, E15. Friedr. Vieweg
   & Sohn, Braunschweig. Translated from the French and edited by Martin Brown from notes by Michel
   Waldschmidt.

S HAFAREVICH , I. R. 1994. Basic algebraic geometry. 1,2. Springer-Verlag, Berlin.

S IEGEL , C. 1948. Analytic functions of several variables. Lectures at the Institute for Advanced Study, 1948
    (reprinted 1962).

TATE , J. T. 1966. Endomorphisms of abelian varieties over finite fields. Invent. Math. 2:134–144.
                                                                                                                163


WATERHOUSE , W. C. 1979. Introduction to affine group schemes, volume 66 of Graduate Texts in Mathemat-
  ics. Springer-Verlag, New York.

W EIL , A. 1948a. Sur les courbes algébriques et les variétés qui s’en déduisent. Actualités Sci. Ind., no. 1041
  = Publ. Inst. Math. Univ. Strasbourg 7 (1945). Hermann et Cie., Paris.

W EIL , A. 1948b. Variétés abéliennes et courbes algébriques. Actualités Sci. Ind., no. 1064 = Publ. Inst. Math.
  Univ. Strasbourg 8 (1946). Hermann & Cie., Paris.

W EIL , A. 1957. On the projective embedding of Abelian varieties, pp. 177–181. In Algebraic geometry and
  topology. A symposium in honor of S. Lefschetz. Princeton University Press, Princeton, N. J.
Index

algebra                                             linear system, 28
     central simple, 79
ample, 30                                           period vectors, 72
     very, 30                                       Poincare sheaf, 37
                                                    polarization, 15, 53
base point, 28                                      polynomial function, 47
birational map, 19                                  principal polarization, 15, 53
                                                    proposition
category                                                 existence of quotients, 38
     of abelian varieties up to isogeny, 78              rigidity lemma, 67
characteristic polynomial, 48
CM-field, 81                                        rational map, 16
CM-type, 82                                         reduction
complex multiplication, 81                               additive, 70
conjecture                                               good, 70, 82
     Mordell’s, 129                                      multiplicative, 70
                                                         semistable, 70, 71
degree                                              representation, 141
     of a polarization, 15, 53                      Riemann form, 13
     of an isogeny, 14, 32                          Rosati involution, 61
direct factor, 64
divisorial correspondence, 37, 86                   seesaw principle, 26
dominating map, 19                                  simple
dual abelian variety, 37                                 abelian variety, 42
dual torus, 15                                           complex torus, 15
dyadic prime, 145                                   subgroup
                                                         arithmetic, 65
fixed divisor, 28                                        congruence, 66
                                                         one-parameter, 10
group
    Neron-Severi, 50                                Tate module, 34
                                                    theorem
height
                                                         Abel, 72
     Faltings, 133
                                                         Chebotarev density, 140
     modular, 133, 156
                                                         de Franchis, 146
     of a normed module, 151
                                                         finiteness II, 132
     stable Faltings, 153
                                                         Hermite, 139
isogeny, 14, 32                                          Jacobi inversion formula, 72
                                                         Mordell-Weil, 69
kernel, 32                                               Neron, 70
Kunneth formula, 12                                      of the cube, 21
                                                         of the square, 23
lattice, 11                                              Riemann hypothesis, 75
Lefschetz trace formula, 2                               rigidity, 8

                                              165
166                                   INDEX


      semisimplicity, 131
      Shafarevich’s conjecture, 132
      Tate’s conjecture, 132
      Wedderburn’s, 79
      Zarhin’s trick, 67

variety, v
     abelian, 8
     group, 7
     Jacobian, 86
     pointed, 86

Weil pairing, 15, 57
Weil q-integer, 79
