Complex Multiplication




       J.S. Milne




        July 14, 2020
    These are preliminary notes for a modern account of the theory of complex multipli-
cation. The article The fundamental theorem of complex multiplication (2007) is a concise
version of part of this work. Probably the two should be read together.




 v0.00 April 7, 2006. First version posted on the web; 113 pages.
 v0.10 July 14, 2020. Updated TeX; minor fixes; 108 pages.                         None

Please send comments and corrections to me at jmilne at umich dot edu.

Available at https://www.jmilne.org/math/CourseNotes/




Copyright c 2006, 2020. J.S. Milne.
Contents

Contents                                                                                       3

I     Analytic Theory                                                                           8
      1     CM-algebras and CM-types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
    Review of semisimple algebras and their modules 8; CM-algebras 10; CM-types. 11; The
    reflex field of a CM-pair 13; The reflex norm. 14; Classification of the primitive CM-pairs
    18; Positive involutions and CM-algebras 19
      2     Complex abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
    Complex tori 21; The cohomology of complex tori 23; Hermitian forms and alternating
    forms 24; Riemann forms 25; Abelian varieties 26
      3     Abelian varieties with complex multiplication . . . . . . . . . . . . . . . . . . 27
    Definition of CM abelian varieties 27; The reflex field of an abelian variety with complex
    multiplication 29; Classification up to isogeny 29; Classification up to isomorphism 31
      4     Mumford-Tate groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
    Review of algebraic groups of multiplicative type 31; CM-pairs and tori 34; The reflex
    norm in terms of tori 35; Complex multiplication in terms of tori 35; Mumford-Tate
    groups 35; Infinity types 37; The Serre group 40 ; Abelian varieties of CM-type 44
      5     Motives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
    The Hodge structure of an abelian variety 45; Abelian motives 45; Hodge structures 45;
    CM-motives 45

II The Arithmetic Theory                                                                     46
   6     Abelian varieties and their good reductions . . . . . . . . . . . . . . . . . . . 46
  Complex abelian varieties and complex tori 46; Specialization of abelian varieties 48; The
  good reduction of abelian varieties 48
   7     Abelian varieties with complex multiplication . . . . . . . . . . . . . . . . . . 52
  Definition of a CM abelian variety 52; Complex multiplication by a Q-algebra 52; Specialization
  54; Rigidity 54; Good reduction 55; The degrees of isogenies 55 ; a-multiplications: first
  approach 58; a-multiplications: second approach 62; a-multiplications: complements 64
   8     The Shimura-Taniyama formula . . . . . . . . . . . . . . . . . . . . . . . . . 66
  Review of numerical norms 66; Statement and proof 66; Alternative approach using
  schemes (Giraud 1968) 70; Alternative approach using p-divisible groups (Tate 1968)
  71 ; Alternative approach using crystals (Deligne c1968) 73; Alternative approach using
  Hodge-Tate decompositions (Serre 1968) 73
   9     The fundamental theorem over the reflex field . . . . . . . . . . . . . . . . . . 74
  Review of the reflex norm 74; Preliminaries from algebraic number theory 74; The funda-
  mental theorem in terms of ideals 75; More preliminaries from algebraic number theory


                                               3
CONTENTS                                                                            4


  76; The fundamental theorem in terms of idèles 77; The fundamental theorem in terms
  of uniformizations 80; The fundamental theorem in terms of moduli 81 ; Alternative
  approach using crystals (Deligne c1968) 83
    10 The fundamental theorem of complex multiplication . . . . . . . . . . . . . . 88
  Statement of the Theorem 88; Definition of f˚ . / 90; Proof of Theorem 10.2 up to an
  element of order 2 93; Completion of the proof (following Deligne) 95

III CM-Motives                                                                           97

IV Applications                                                                          98

A Additional Notes; Solutions to the Exercises                                           99

B Summary                                                                                101

Bibliography                                                                             104

Index of definitions                                                                     107
Preface

Abelian varieties with complex multiplication1 are special in that they have the largest pos-
sible endomorphism rings. For example, the endomorphism ring of an elliptic curve is
usually Z, but when it is not, it is an order in an imaginary quadratic number field, and
the elliptic curve is then said to have complex multiplication. Similarly, the endomorphism
ring of a simple abelian variety of dimension g is usually Z, but, at the opposite extreme, it
may be an order in a number field of degree 2g, in which case the abelian variety is said to
have complex multiplication. Abelian varieties with complex multiplication correspond to
special points on the moduli variety of abelian varieties, and their arithmetic is intimately
related to that of the values of modular functions and modular forms at those points.
     The first important result in the subject, which goes back to Kronecker and Weber,
states that the Hilbert class field (maximal abelian unramified extension) of an imaginary
quadratic subfield E of C is generated by the special value j. / of the j -function at any
element  of E in the complex upper half plane generating the ring of integers in E. Here
j is the holomorphic function on the complex upper half plane    p
                                                                    invariant under the action
                                                                           p
of SL2 .Z/, taking the values 0 and 1728 respectively at 1C2 3 and             1, and having a
simple pole at infinity. The statement is related to elliptic curves through the ideal class
group of E, which acts naturally both on the Hilbert class field of E and on the set of
isomorphism classes of elliptic curves with endomorphism ring OE .
     Generalizing this, Hilbert asked in the twelfth of his famous problems whether there ex-
ist holomorphic functions whose special values generate the abelian extensions (in particu-
lar, the class fields) of arbitrary number fields. For imaginary quadratic fields, the theory of
elliptic curves with complex multiplication shows that elliptic modular functions have this
property (Kronecker, Weber, Takagi, Hasse). Hecke began the study of abelian surfaces
with complex multiplication in the early 1900s, but the primitive state of algebraic geome-
try over fields other than C made this premature. It was not until the 1950s, after Weil had
developed the theory of abelian varieties in arbitrary characteristic, that he, Shimura, and
Taniyama were able to successfully extend the main statements of the theory of complex
multiplication from elliptic curves to abelian varieties. While the resulting theory has pro-
vided only a partial answer to Hilbert’s problem, it has played an essential role in the theory
of modular (and, more generally, Shimura) varieties and in other aspects of number theory.
     The complex points of a modular variety parametrize polarized abelian varieties over
C together with a level structure; at a special point, the abelian variety has complex multi-
plication. To understand the arithmetic nature of the values of modular functions at these
special points, it is necessary to understand how abelian varieties with complex multiplica-
    1 The name is both archaic and imprecise — the term “multiplication” is no longer used to denote an

endomorphism, and “complex multiplication” is sometimes used to denote a more general class (Birkenhake
and Lange 2004, p. 262) — but I know of no other.



                                                  5
CONTENTS                                                                                                   6


tion and their torsion points behave under automorphisms of C (as an abstract field). For
automorphisms of C fixing a certain “reflex” field attached to the abelian variety, this is the
main content of the theory of Shimura, Taniyama, and Weil from the 1950s. Their results
were extended to all automorphisms of C by the later work of Deligne, Langlands, and Tate.

N OTATION .
By a field we always mean a commutative field. A number field is a field of finite degree
over Q.2 An algebraic closure of a field k is denoted k al . We let C denote an algebraic
closure of R and Qal the algebraic closure of Q in C. We often use Q  N to denote an algebraic
                                   al
closure of Q (not necessarily Q ). Complex conjugation on C (or a subfield) is denoted
by  or simply by a 7! Na. A complex conjugation on a field k is an involution induced by
complex conjugation on C and an embedding of k into C.3 An automorphism  of a field
˝ is said to fix a subfield k if  a D a for all a 2 k.
    When k is a field, an étale algebra over k is a finite product of finite separable field
extensions of k. Let E be an étale Q-algebra, and let k be a field containing Q. We say that
k contains all conjugates of E if every Q-algebra homomorphism E ! k al maps into k;
equivalently, if there are ŒEW Q distinct Q-algebra homomorphisms E ! k.
    Rings are assumed to have a 1, homomorphisms of rings are required to map 1 to 1,
and 1 acts on every module as the identity map. By a k-algebra (k a field) I mean a ring B
containing k in its centre.
    Following Bourbaki, I require compact topological spaces to be separated (Hausdorff).
    An algebraic variety over a field k is a geometrically reduced scheme of finite type over
k. If V and V 0 are algebraic varieties over a field k, then a morphism V ! V 0 means
a morphism (regular map) defined over k. If K is a field containing k, then VK is the
algebraic variety over K obtained by extension of the base field and V .K/ is the set of
points of V with coordinates in K.4 If  W k ,! K is a homomorphism of fields and V is an
algebraic variety (or other algebro-geometric object) over k, then V has its only possible
meaning: apply  to the coefficients of the equations defining V . The tangent space at a
point P of a variety V is denoted by TgtP .V /.
    Let A and B be sets and let  be an equivalence relation on A. If there exists a canonical
surjection A ! B whose fibres are the equivalence classes, then I say that B classifies the
elements of A modulo  or that it classifies the -classes of elements of A.

R EFERENCES
In addition to those listed at the end, I refer to the following of my books and course notes
(the latter are available at www.jmilne.org/math/).
iAG: Algebraic Groups, Cambridge U.P., 2017.
AG: Algebraic Geometry, Chapter 16. Descent Theory 2015.
ANT: Algebraic Number Theory, v3.07, March 2017.
CFT: Class Field Theory, v4.02, March 2013.
    2 Following Kronecker (see Vlüaduţ 1991, p. 12), we do not assume F to be a subfield of C.
    3 More precisely, it is an automorphism 0 of order 2 of k such that  ı 0 D  ı  for some homomorphism

W k ! C. Thus, a complex conjugation on k is defined by a homomorphism W k ! C such that .k/ is stable
under  but not fixed by it. The complex conjugations on C are the conjugates of  by automorphisms of C.
According to a theorem of Artin (Collected Papers p. 257), the complex conjugations on Q  N are exactly the
                  N
elements of Gal.Q=Q/ of order 2.
     4 In other words, V .K/ D Mor
                                  Spec.k/ .Spec.K/; V /.
CONTENTS                                                                             7


FT: Fields and Galois Theory, v4.61, April 2020.

P REREQUISITES
The reader is expected to have a good knowledge of basic algebraic number theory (e.g.,
ANT and parts of CFT), and basic algebraic geometry (e.g., AG and Hartshorne 1977, II)
including abelian varieties (e.g., Milne 1986).
Chapter I

Analytic Theory

1    CM-algebras and CM-types
Review of semisimple algebras and their modules
Fix a field k of characteristic zero. In this subsection, all k-algebras B, and all B-modules,
will be of finite dimension over k.
     A k-algebra is said to be semisimple if it has no nonzero nilpotent ideals1 , and it is said
to be simple if it has no nonzero two-sided ideals. The Wedderburn theorems state that a
semisimple k-algebra is a direct product of its minimal two-sided ideals, each of which is a
simple k-algebra; moreover, each simple k-algebra is isomorphic to a matrix algebra over a
division k-algebra. For example, a commutative semisimple k-algebra is simply a product
of fields.
     We now describe the modules over a semisimple k-algebra B. Every such module is
semisimple, and hence a direct sum of simple modules. Thus, it suffices to describe the
simple modules. Suppose B  Mn .D/, and choose an isomorphism; then D n becomes a
B-module under  Q left multiplication; it is simple, and every simple B-module is isomorphic
to it. Let B D 1i n Bi be the decomposition of B into a product of its simple ideals, and
let Si be a simple Bi -module. When we let B act on Si through the projection B ! Bi ,
each Si becomes a simple L B-module, and every     L B-module L is isomorphic to a direct sum of
copies of the Si , S  niD1 ri Si ; moreover, niD1 ri Si  niD1 ri0 Si if and only if ri D ri0
for all i .
                                                                               def
     Let k 0 be a field containing k. If B is semisimple, then so also is B 0 D B ˝k k 0 (here
is where we use that k has characteristic zero), but the analogous statement with “simple”
is false. Consider, for example, a simple Q-algebra B, and let k be its centre. Then k is a
field, and for a field K containing all conjugates of k and splitting B,
                                                     Y
            a ˝ c $ .: : : ; .a/c; : : :/W k ˝Q K '          K
                                                       Wk!K
                                                     Y
                                           B ˝Q K            Mn .K/; n2 D ŒBW k:
                                                               Wk!K

Let V D K n be the simple
                     L B ˝Q K-module corresponding to W k ! K. Any B ˝Q K-
module isomorphic to  V is said to be reduced.

    1 An ideal a is nilpotent if ar D 0 for some r. In particular, its elements are nilpotent.




                                                        8
CHAPTER I. ANALYTIC THEORY                                                                                       9


P ROPOSITION 1.1 Let B be a simple Q-algebra with centre k. Let K be a field containing
all conjugates of k and splitting B, and let V be a B ˝Q K-module. The characteristic
polynomials det.T bjV / of all elements b of B acting on the K-vector space V have
coefficients in Q if and only if V is isomorphic to a multiple of the reduced B ˝Q K-module
(equivalently, V is free as a k ˝Q K-module).
                                            Q
P ROOF. The isomorphism   Q ˛W B ˝Q K ! Wk!K Mn .K/ is well-determined up to conju-
gation by an element of Wk!K Mn .K/ , and hence the characteristic polynomial Pb .T /
of ˛.b ˝ 1/ for b 2 B is well-defined. It equals det.T bjV /, where V is the reduced mod-
ule. For any automorphism  of K,  .Pb .T // D Pb .T /. Enlarging K to a Galois extension
of Q doesn’t change Pb .T /, and so this shows that Pb .T / has coefficients
                                                                        L      in Q.
     Any other B ˝Q K-module M is isomorphic to a direct sum  m V , m  0: As
L2 k acts on VQ as multiplication
a
                             m
                                 n by .a/, the characteristic polynomial PM;a .T / of a on
     m V is     .T   a/       . When a generates k, this has coefficients in Q if and only
                               

if the m are equal. 2
                                                                                              2
                                                     Q
     Let B be a semisimple k-algebra, and let B D Bi be its decomposition into a product
of simple algebras Bi . The centre of each Bi is a field ki , and each degree ŒBi W ki  is a
square. The reduced degree of B over k is defined to be
                                          X
                              ŒBW kred D     ŒBi W ki 1=2  Œki W k:
                                                         i

For any field k 0 containing k, ŒBW k D ŒB ˝k k 0 W k 0  and

                                        ŒBW kred D ŒB ˝k k 0 W k 0 red :                                     (1)

P ROPOSITION 1.2 Let B be a semisimple k-algebra. For any faithful B-module M ,

                                            dimk M  ŒBW kred ;

and there exists a faithful module for which equality holds if and only if the simple factors
of B are matrix algebras over their centres.
                    Q                    Q
P ROOF. Let B D Bi , where Bi  Mni .Di / with Di a central division algebra over
                    ni
      L let Si D Di be a simple Bi -module. Then every B-module M is isomorphic to a
ki , and
sum i mi Si , and M is faithful if and only if each mi > 0. Therefore, if M is faithful,
                          X                                   X
              dimk M D        mi  ni  ŒDi W k  Œki W k    ni  ŒDi W k  Œki W k:
                                   i                                        i

On the other hand,                               X                     1
                                  ŒBW kred D            ni  ŒDi W k 2  Œki W k:
                                                     i
The proposition is now obvious.                                                                                  2

P ROPOSITION 1.3 Let B be a semisimple k-algebra. Every maximal étale k-subalgebra of
B has degree ŒBW kred over k.
P ROOF. When B is central simple, the proposition asserts that every maximal subfield of
                                 1
B containing k has degree ŒBW k 2 . This case is proved in CFT, IV 3.5, and the general case
follows easily.                                                                             2
    2 Let c .T / D
                     Q
           a            .T   a/ be the characteristic polynomial of a in the field extension k=Q. Because a
generates k, ca .T / is irreducible. Any monic irreducible factor of PM;a .T / in QŒT  shares a root with ca .T /,
and therefore equals it. Hence, if PM;a .T / has coefficients in Q, it is a power of ca .T /.
CHAPTER I. ANALYTIC THEORY                                                                    10


CM-algebras
A number field E is said to be totally real if its image under every homomorphism E ,! C
is contained in R. When the image is never contained in R, the field is said to be totally
imaginary. Equivalently, E is totally real if E ˝Q R  RŒE WQ and it is totally imaginary if
E ˝Q R  CŒE WQ=2 . A number field QŒ˛ ' QŒX =.f .X // is totally real if all the roots of
f .X / are real and it is totally imaginary if none of the roots are real.

P ROPOSITION 1.4 The following conditions on a number field E are equivalent:
  (a) E is a totally imaginary quadratic extension of a totally real number field;
  (b) there exists a nontrivial automorphism E of E such that  ı E D  ı  for all homo-
      morphisms W E ,! C;
  (c) E D F Œ˛ with F totally real, ˛ 2 2 F , and .˛ 2 / < 0 for all homomorphism W F ,!
      C.

P ROOF. Assume (a), and let F be the totally real subfield. The unique nontrivial auto-
morphism of E fixing F has the property required for (b). Let ˛ generate E over F . After
completing the square, we may suppose that ˛ 2 2 F , and then .˛ 2 / < 0 for any embedding
W F ,! C because otherwise .E/  R.
     Assume (b). Then E has order 2, because  ı 2E D  for every W E ,! C. Moreover,
its fixed field F is totally real, E is quadratic over F , and E is totally imaginary (because 
acts nontrivially on .E/ for all ).
     Assume (c). Certainly, the conditions imply that E is a totally imaginary quadratic
extension of F .                                                                              2

   Because they occur in the theory of complex multiplication, the fields satisfying these
conditions are called CM-fields. Note that a number field E is CM if and only if it has
exactly one complex conjugation (by (b)). Clearly, every field isomorphic to a CM-field is
CM.

C OROLLARY 1.5 A finite composite of CM-subfields of a field is CM; in particular, the
Galois closure of a CM-field in any larger field is CM.

P ROOF. Clearly, each complex embedding of the composite of two CM-fields will induce
the same nontrivial complex conjugation on the field.                               2

R EMARK 1.6 Let K  Qal be a number field. If   1 acts on K as  for every  2
Aut.Qal /, then K is totally real or is a CM-field according as  fixes E or not. It follows that
                                                                                    def
the union of all CM-subfields of Qal is the field fixed by the commutators Œ;  D   1  1
of Gal.Qal =Q/, i.e., it is the subfield corresponding to the closure of the group generated by

                                  fŒ;  j  2 Gal.Qal =Q/g:

We denote this field by Qcm .

R EMARK 1.7 Let K be a number field. Since a composite of totally real fields is totally
real, K contains a largest totally real subfield F . Moreover, K contains at most one totally
                                                                                        p
imaginary quadratic extensionpof F , because every such extension is of the form F Œ ˛
with ˛ totallypnegative; if F Œ ˇ is a second such extension, then K containspthe totally
                                                                      p
real field F Œ ˛ˇ, which must equal F , and this implies that F Œ ˛ D F Œ ˇ. If K
CHAPTER I. ANALYTIC THEORY                                                                   11

                                  def
contains a CM-field E, then K 0 D E  F is the largest CM-subfield of K. It consists of all
elements ˛ of K having a conjugate ˛ 0 in K such that .˛ 0 / D .˛/ for all embeddings
W K ! C. For any such embedding, K 0 D K \ Qcm .

     A CM-algebra is a finite product of CM-fields. Equivalently, it is a finite product of
number fields admitting an automorphism E that is nontrivial on each factor and such that
 ı  D  ı E for all homomorphisms W E ! C. The fixed algebra of E is a product of the
largest totally real subfields of the factors. Sometimes we call E complex conjugation and
write aN for E a.

CM-types.
Let E be a CM-algebra. The Q-algebra homomorphisms E ! C occur in complex con-
jugate pairs f';  ı 'g. A CM-type on E is the choice of one element from each such pair.
More formally, we have the following definition.

D EFINITION 1.8 A CM-type on a CM-algebra is a subset ˚  Hom.E; C/ such that
                                                                    def
             Hom.E; C/ D ˚ t ˚              (disjoint union; ˚ D f ı ' j ' 2 ˚g).

    Alternatively, we may regard a CM-type as a function W Hom.E; C/ ! f0; 1g (the char-
acteristic function of ˚) such that

                         ./ C . ı / D 1 for all  2 Hom.E; C/:                         (2)

Let F be the product of the largest totally real subfields of the factors of E. Choosing a
CM-type ˚ on E amounts to choosing an extension 0 to E of each embedding W F ! R,
and hence an isomorphism of R-algebras
                    ˚   Y
         E ˝Q R !                    C;    a ˝ r 7! .0 a  r/ ;   ˚ D f0 j W F ! Rg:    (3)
                            WF !R

A pair .E; ˚/ (or .E; /) consisting of a CM-algebra E and a CM-type ˚ (or ) for E will
be called a CM-pair.
    Let E0 be a CM-subalgebra of a CM-algebra E. Every CM-type ˚0 on E0 , extends to
a CM-type on E, namely, to
                                     def
                               ˚ D f'W E ! C j 'jE0 2 ˚0 g;

and a CM-type ˚ on E arises in this way from a CM-type on E0 if and only if
                                            def
                                     ˚jE0 D f'jE0 j ' 2 ˚g

is a CM-type on E0 , i.e., no two of the ' in ˚ become complex conjugates on E0 (or, if
two elements of ˚ have the same restriction to F0 , then they have the same restriction to
E0 ).
    A CM-pair .E; ˚ /, or just ˚ itself, is primitive if E is a field and there does not exist a
proper CM-subfield E0 of E such that ˚jE0 is a CM-type on E0 .
CHAPTER I. ANALYTIC THEORY                                                                     12


P ROPOSITION 1.9 Every CM-pair .E; ˚ / with E a field is the extension of a unique prim-
itive CM-pair .E0 ; ˚0 / with E0  E. In fact, for any CM-field E1 containing E and Galois
over Q, E0 is the fixed field of

                            H D f 2 Gal.E1 =Q/ j ˚1  D ˚1 g:

Here ˚1 is the extension of ˚0 to E1 and ˚1  D f' ı  j ' 2 ˚1 g.

P ROOF. Assume initially that E is Galois over Q, and define E0 to be the fixed subfield of
H D f 2 Gal.E=Q/ j ˚  D ˚g:
(A) E0 is a CM-subfield of E and ˚jE0 is a CM-type on E0 .
    As E is a CM-field,
                                    ˚ E D ˚ ¤ ˚;
and so E is not in H ; it therefore acts nontrivially on E0 . To show that E0 is a CM-
subfield, it remains to show that it is stable under E , i.e., that  E a D E a for all  2 H
and a 2 E0 . But, for  2 H ,

                              ˚ E  E D ˚  E D ˚ E D ˚;

and so E E 2 H . This implies that  E a D E a for all a 2 E0 .
   If ˚jE0 is not a CM-type, then

                                       ' 0 jE0 D  ı 'jE0                                      (4)

for distinct '; ' 0 2 ˚. But (4) implies that  ı ' 2 ' 0 H  ˚, which is a contradiction.
(B) If E 0 is a CM-subfield of E and ˚jE 0 is a CM-type on E 0 , then E 0  E0 .
                                                                                def
     The conditions imply that ˚ is the extension to E of the CM-type ˚ 0 D ˚jE 0 on E 0 .
Let  be an element of G fixing E 0 . Then ˚ 0  D ˚ 0 , which implies that ˚  D ˚, and so
 2 H.
     (A) and (B) prove the proposition when E is Galois over Q. In the general case, we
can embed E in a CM-field E1 Galois over Q and extend ˚ to a CM-type ˚1 on E1 . The
preceding argument applied to .E1 ; ˚1 / gives a smallest CM-field E0  E such that ˚jE0
is a CM-type on E0 .                                                                       2

C OROLLARY 1.10 A CM-pair .E; ˚ / is primitive if and only if for some (hence all) CM-
fields E1 containing E and Galois over Q, the subgroup of Gal.E1 =Q/ fixing E is

                               f 2 Gal.E1 =Q/ j ˚1  D ˚1 g;

where ˚1 is the extension of ˚ to E1 .

P ROOF. Immediate from the proposition.                                                         2

E XERCISE 1.11 (Shimura and Taniyama 1961, 8.2 = Shimura 1998, 8.2). Let E be a CM-
field, and write E D F Œ˛ with ˛ 2 2 F and totally negative. The embeddings 'W E ! C
such that =.'.˛// > 0 form a CM-type ˚ on E. Show that .E; ˚ / is primitive if and only
if
   (a) F Œ˛ D QŒ˛, and
  (b) for any conjugate ˛ 0 of ˛ over Q other than ˛ itself, ˛ 0 =˛ is not totally positive.
CHAPTER I. ANALYTIC THEORY                                                                                         13


E XERCISE 1.12 (ibid. 8.4). Let E D QŒ, where  is a primitive 13th root of 1 in C.
Of the 32 CM-types on E containing the given embedding of E into C, show that only 2
are nonprimitive, and that the remaining 30 CM-types fall into 6 orbits under the action of
Gal.E=Q/, each with 5 elements.
                                                       N be an algebraic closure of Q. A
D EFINITION 1.13 Let E be an étale Q-algebra, and let Q
CM-type on E with values in N
                            Q is a subset ˚ of HomQ-alg .E; N
                                                            Q/ such that

                                                       N D ˚ t ˚
                                          HomQ-alg .E; Q/

                                  N
for all complex conjugations  on Q.

     Note that when E is a CM-algebra and N
                                          Q D Qal , this agrees with Definition 1.8.

E XERCISE 1.14 Let ˚ be a CM-type on a field E with values in Q.N Show that there exists
a CM-subfield E0 of E such that no two elements of ˚ are complex conjugates on E0 (and
hence there is a CM-type ˚0 on E0 such that ˚ D f'W E ! Q N j 'jE0 2 ˚0 g).

E XERCISE 1.15 Rewrite this subsection replacing Qal and C with N
                                                                Q.

The reflex field of a CM-pair
If  is an automorphism of C (or Qal / and ˚ is a CM-type on a CM-algebra E, then
                                                def
                                             ˚ D f ı ' j ' 2 ˚g

is again a CM-type on E.3

P ROPOSITION 1.16 Let .E; ˚ / be a CM-pair. The following conditions on a subfield E 
of Qal are equivalent:
  (a)  2 Gal.Qal =Q/ fixes E  if and only if  ˚ D ˚ ;
  (b) E  is the subfield of N
                                                         P
                             Q generated by the elements   '.a/, a 2 E.      '2˚

                    al
     P. If  2 Gal.Q =Q/ permutes the '’s in ˚, then clearly it fixes all elements of the
P ROOF
form '2˚ '.a/. Conversely, if
                                X              X
                                      '.a/ D          . ı '/.a/ for all a 2 E  ;
                                '2˚            '2˚

then f ı ' j ' 2 ˚g D ˚ by Dedekind’s theorem on the independence of characters (FT
5.14).4 This shows that conditions (a) and (b) define the same field.              2

D EFINITION 1.17 The field satisfying the equivalent conditions in the proposition is called
the reflex field E  of .E; ˚ /.

     Note that, in contrast to E, which need not even be a field, E  is a subfield of Qal .
    3 Note that, because E is CM,  ı . ı '/ D . ı '/ ı                                                      0
                                                              E D  ı . ı '/; therefore, if  ı . ı '/ D  ı ' , then
 ı . ı '/ D  ı ' 0 and  ı ' D ' 0 .
                                      Hence ˚ \ ˚ D ;, and it follows (by counting) that Hom.E; C/ D
˚ t  ˚.
    4 In more detail, the equation says that
                                             P         P
                                               '2 ˚ '   '2˚ ' D 0, and Dedekind’s theorem says that this is
possible only if each ' in ˚ occurs exactly once in ˚.
CHAPTER I. ANALYTIC THEORY                                                                14


P ROPOSITION 1.18 Let .E; ˚ / be a CM-pair.
  (a) The reflex field E  of .E; ˚ / is a CM-field.
  (b) If .E; ˚/ D 1i m .Ei ; ˚i /, then E  D E1    Em
                                                           :
                   Q

  (c) The reflex field of any extension .E1 ; ˚1 / of .E; ˚ / equals that of .E; ˚ /.

P ROOF. (a) Let  2 Gal.Qal =Q/ and a 2 E. Because E is a CM-algebra,
                          P                P             
                            '2˚ '.a/ D       '2˚ '.E a/
                                           P
                                         D '2˚ . ı '/.E a/
                                            P                
                                         D    '2˚ . ı '/.a/
                                              P          
                                         D     '2˚ '.a/   ;

and so E  is either CM or totally real (cf. 1.6). As ˚ ¤ ˚, it must be CM.
   (b) Because ˚ D ˚1 t : : : t ˚n as Gal.Qal =Q/-sets,
                                               \
                          f j  ˚ D ˚g D          f j  ˚i D ˚i g:
                                                            i

     (c) Clearly ˚1 D . ˚ /1 , and so  ˚1 D ˚1 ”  ˚ D ˚.                               2

E XAMPLE 1.19 Consider a CM-pair .E; ˚ / with E a subfield of Qal . Let E1 be the Galois
closure of E in Qal , and let ˚1 be the extension of ˚ to E1 . Regard the elements of ˚1
as automorphisms of E1 , and let ˚1 1 D f' 1 j ' 2 ˚1 g. Then ˚1 1 is a CM-type on
E1 , and the primitive subpair .E0 ; ˚0 / of .E1 ; ˚1 1 / (see 1.9) has E0 D E  , the reflex
field of .E; ˚/.5 The pair .E0 ; ˚0 / is denoted .E  ; ˚  / and called the reflex CM-pairof
.E  C; ˚/ (and ˚  is called the reflex CM-type of ˚).

E XERCISE 1.20 Rewrite this subsection replacing Qal and C with N
                                                                Q. Is it necessary to
assume that E is a CM-algebra?

The reflex norm.
In this subsection, N
                    Q is an algebraic closure of Q (not necessarily the algebraic closure in
C). By a CM-pair we mean a CM-algebra E together with a subset ˚  Hom.E; Q/        N such
that
                                   Hom.E; N Q/ D˚ t 0 ˚
                                           0    N                   
                                        P  on Q. The reflex field E of .E; ˚ / is the
for one (hence every) complex conjugation
            N generated by the elements
subfield of Q                            '2˚ '.a/ with a 2 E, and

                                           Q=E  / D f j  ˚ D ˚g:
                                       Gal. N
    5 Recall (1.9) that E is the fixed field of the group
                         0

                                               f j ˚1 1  D ˚1 1 g;

and (1.16) that E  is the fixed field of the group

                                        f j ˚ D ˚g D f j ˚1 D ˚1 g:

Obviously, these groups are equal.
CHAPTER I. ANALYTIC THEORY                                                              15


     Let k be a number field. To give a finitely generated E ˝Q k-module amounts to giving
a finite-dimensional Q-vector space together with commuting Q-linear actions of E and k
(i.e., an .E; k/-bimodule over Q), or a finite-dimensional k-vector space V together with
a k-linear action of E (i.e., an action of E on V such that each a 2 E acts by k-linear
endomorphisms).

P ROPOSITION 1.21 Let .E; ˚ / be a CM-pair, and let k be a subfield of N
                                                                       Q. There exists a
finitely generated E ˝Q k-module V such that
                                        X
                           Trk .ajV / D   '.a/; all a 2 E;                           (5)
                                               '2˚

if and only if k  E  , in which case V is uniquely determined up to an E ˝Q k-isomorphism.

P ROOF. If a acts k-linearly on V , then Trk .ajV / 2 k, and so, if there exists a k-linear
action of E satisfying (5) on a k-vector space V , then certainly k  E  .
    For the converse, we initially assume that k contains all the conjugates of E. There is
then a canonical isomorphism
                                                        Y
                         e ˝ a 7! .e  a/ W E ˝Q k !             k;
                                                                        WE !k
                                                      L
and so any E ˝Q k-module V is of the form m k for unique nonnegative integers m ,
where k denotes a one-dimensional k-vector space on which e 2 E acts as .e/.LThus, up
to isomorphism, there exists exactly one E ˝Q k-module satisfying (5), namely, '2˚ k' .
     For a general k containing E  , we use the following statement:
       Let ˝ be a finite Galois extension of k with Galois group ; the functor V
       ˝ ˝k V is an equivalence from the category of k-vector spaces to the category
       of ˝-vector spaces endowed with a semilinear action of (see AG 16.14; an
       action is semilinear if .av/ D a  v for 2 , a 2 ˝, v 2 V ).
Let ˝ be any finite L  Galois extension ˝ of k containing all conjugates of E. Consider
the E ˝Q ˝-module '2˚ ˝' , where ˝' is a one-dimensional ˝-vector space on which
e 2L E acts as '.e/. Because ˚ is stable under , we can define a semilinear action of
on '2˚ ˝' by the rule
                                   '                     ı'
                            .: : : ; v; : : :/ D .: : : ; v ; : : :/;     2 ;

and one checks that this is the only such action commuting
                                                       Q     with the action of E.6 Any
E ˝Q k-module satisfying (5) becomes isomorphic to '2˚ ˝' over ˝, and so this shows
that, up to isomorphism, there exists exactly one such E ˝Q k-module.                 2

                                                                 N such that there exists
C OROLLARY 1.22 The reflex field E  is the smallest subfield of Q
         
an E ˝Q E -module V with
                           M
                      N '
               V ˝E  Q             QN'                    N
                                             (as an E ˝Q Q-module),                   (6)
                                       '2˚

where N
      Q' is a one-dimensional N
                              Q-vector space on which E acts through '.
   6 Use that

                             ˝' D fx 2 V j a  x D '.a/x all a 2 Eg:
CHAPTER I. ANALYTIC THEORY                                                              16


P ROOF. Restatement of the proposition.                                                  2

    Let V˚ be an E ˝Q k-module satisfying (5). An element a of k defines an endomor-
phism of V˚ regarded as an E-vector space, whose determinant we denote by detE .ajV˚ /.
If a 2 k  , then detE .ajV˚ / 2 E  , and so in this way we get a homomorphism

                                    Nk;˚ W k  ! E  :

More generally, for any Q-algebra R and invertible element of a of k ˝Q R, we get an
invertible element
                          Nk;˚ .a/ D detE ˝Q R .ajV˚ ˝Q R/
of E ˝Q R. In this way, we get a homomorphism

                           Nk;˚ .R/W .k ˝Q R/ ! .E ˝Q R/

which is functorial in R and independent of the choice of V˚ . It is called the reflex norm
from k to E (relative to ˚). When k D E  , we drop it from the notation.
                                                        N
P ROPOSITION 1.23 For any number field k with E   k  Q,

                                 Nk;˚ D N˚ ı Nmk=E  :                                  (7)

P ROOF. Choose an E ˝Q E  -module V˚ satisfying (6), and let V 0 D k ˝E  V˚ . When we
use V 0 to compute Nk;˚ , and V˚ to compute N˚ , we obtain (7).                       2

R EMARK 1.24 (a) For any isomorphism  W E ! E 0 ,

                             N˚  .a/ D N˚ .a/, all a 2 E  ,

where ˚ D f' ı  j ' 2 ˚g.
   (b) Let V˚ be an E ˝Q k-module satisfying (5). Then V˚ ˚ V˚ satisfies
                                     X
                        Trk .ajV / D        N
                                              .a/, all a 2 E:
                                           WE !Q

Therefore V˚ ˚ V˚ is a free E ˝Q k-module of rank 1, and so

                                       N˚ .a/  N˚ .a/

Therefore V˚ ˚ V˚ is free of rank 1, and so

                        N˚ .a/  N˚ .a/ D Nmk=Q .a/, all a 2 k  .                     (8)

Since N˚ .a/ D N˚ E .a/ D E N˚ .a/, this can be rewritten as

                       N˚ .a/  E N˚ .a/ D Nmk=Q .a/, all a 2 k  :                    (9)

More generally, for any Q-algebra R,

                 N˚ .a/  E N˚ .a/ D Nmk˝Q R=R .a/, all a 2 .k ˝Q R/ :              (10)
CHAPTER I. ANALYTIC THEORY                                                                  17


R EMARK 1.25 In terms of algebraic tori (see 4), Nk;˚ is a homomorphism T k ! T E ,
where T k and T E are the algebraic tori over Q with Q-points k  and E  respectively (i.e.,
T k D .Gm /k=Q and T E D .Gm /E=Q ).
    Let F be the largest totally real subalgebra of E. The norm a 7! a  E aW E  ! F 
defines a homomorphism T E ! T F , and we let T equal the fibre product T D Gm T F
T E:
                                       T         TE


                                           Gm          TF:

Thus T is the subtorus of T E with

                                T .Q/ D fa 2 E  j a  iE a 2 F  g:

Equation (10) shows that the homomorphism Nk;˚ W T k ! T E factors through T  T E .
    From Nk;˚ we obtain homomorphisms (by taking R D Q; Ql ; R):

                     N0 W k  ! E  ;
                        Nl W kl ! El ;   kl D k ˝Q Ql ;    El D E ˝Q Ql ;
                              
                    N1 W k1 ! E1 ;         k1 D k ˝Q R;      E1 D E ˝ R:

From these maps, we get a continuous homomorphism on the groups of idèles A   k
                                                                                  ! A E
which is compatible with N0 , and hence induces a homomorphism on the idèle classes.
Moreover, the homomorphism on the finite idèles passes to the quotient and defines a ho-
momorphism Nk;˚ on the groups of fractional ideals, which is compatible with N0 , and so
induces a homomorphism on the ideal classes.
P ROPOSITION 1.26 Let k  Q  N be a finite extension of E  containing all conjugates of E.
For any nonzero element or fractional ideal a of k,
                                       Y
                         Nk;˚ .a/ D            ' 1 .Nmk='E a/:                         (11)
                                                '2˚

P ROOF. For a 2 k  ,
                             detE .aW k' ! k' / D ' 1 .Nmk='E a/;
which implies (11) in this case.
    Each side of (11) defines a homomorphism on the groups of fractional ideals, which
are torsion-free, and so it suffices to prove that the two homomorphisms agree on principal
ideals. This we have just done.                                                          2

                                             N of E  containing all conjugates of E
C OROLLARY 1.27 For any finite extension k  Q
and fractional ideal a of E  ,
                                        Y
                          N˚ .a/ŒkWE  D    ' 1 .Nmk='E a0 /;                    (12)
                                                 '2˚

where a0 is the extension of a to a fractional ideal of k (so a0 D aOk if a is integral).
P ROOF. We have
                                     .7/                               
                          Nk;˚ .a0 / D N˚ .Nmk=E  a/ D N˚ .aŒkWE  /
and so (12) follows from (11).                                                              2
CHAPTER I. ANALYTIC THEORY                                                                       18


    Note that (12) determines N˚ as a homomorphism from the fractional ideals of E  to
the fractional ideals of E.
E XAMPLE 1.28 Consider a CM-pair .E; ˚ / with E a subfield of Q. N The reflex CM-pair
.E ; ˚ / can be described as follows: choose a subfield L of N
                                                           Q containing E and Galois
                         N
over Q (for example, L D Q) and let ˚L D f 2 Gal.L=Q/ j jE 2 ˚g; then

                           Gal.L=E  / D f 2 Gal.L=Q/ j  ˚ D ˚gI

the set ˚L is stable under the left action of Gal.L=E  /, and when we write
                             G
                     ˚L 1 D           
                                          Gal.L=E  / (disjoint union),
                                       2˚L

˚  D f jE  j      2 ˚L g is the reflex CM-type on E  . The map
                                       Y
                                 a 7!        
                                                .a/W E  ! L
                                                2˚

factors through E   L , and the resulting map E  ! E  is N˚ — this is a restatement
of (1.26). Because it has this description, other authors write N˚  where we write N˚ .7

Classification of the primitive CM-pairs
An isomorphism of CM-pairs .E; ˚ / ! .E 0 ; ˚ 0 / is an isomorphism ˛W E ! E 0 of Q-
algebras such that ' ı ˛ 2 ˚ whenever ' 2 ˚ 0 .
                                                           N Galois over Q and containing
    Let .E; ˚/ be a CM-pair, and let k be a CM subfield of Q
                N              N
E . For W E ! Q and  2 Gal.Q=Q/, define
                                         
                                            1 if  2  ˚
                                 . / D
                                            0 otherwise.
                                     1 ı /, where  is the characteristic function of ˚.
In other words,      ./ D .

L EMMA 1.29 For each , the number  . / depends only on the restriction of  to E 
and the map
                                             N ! f0; 1g
                          7!  . /W Hom.k; Q/
is a CM-type on k:
P ROOF. If  0 jE  D jE  , then  0 D  ı  for some  fixing E  ,8 and so

                                           0˚ D   ˚ D  ˚ I
             0
hence     . / D     . /. As  ˚ is a CM-type on E,  lies in exactly one of  ˚ or  ˚,
and so
                                          . / C        . ı / D 1:                            2

                    N
    For any  2 Gal.Q=Q/,
                                          1                         1
                       ı . / D .         ı  ı / D      .       ı  / D .    /. /;

                                           N
and so, as  runs over the embeddings E ,! Q,                                        N
                                                             runs over a Gal.Q=Q/-orbit of CM-types
on k.
   7 For us, the reflex CM-type plays almost no role.
   8 Think of  and  0 as automorphisms of k, and take  D  1 ı  0 .
CHAPTER I. ANALYTIC THEORY                                                                                        19


P ROPOSITION 1.30 The map .E; ˚ / 7! f  g defines a bijection from the set of isomor-
phism classes of primitive CM-pairs .E; ˚ / whose reflex field is contained in k to the set
       N
of Gal.Q=Q/-orbits  of CM-types on k.

P ROOF. We construct an inverse. For a CM-type on k, let .E ; ˚ / be the reflex CM-
pair of .k; / (see 1.19). By definition, .E ; ˚ / is the primitive subpair of .k; 1 /.
Its isomorphism class depends only on the Gal.Qal =Q/-orbit of ,9 and the map 7!
.E ; ˚ / provides the required inverse.10                                           2

     Let k be a composite of CM-subfields of N Q (e.g., k could be the composite Qcm of all
                  N We define a CM-type on k to be a locally constant map W Hom.k; Q/
CM-subfields of Q).                                                                  N !
                                                                                 cm
f0; 1g such that ./ C . ı / D 1 for all . For example, the CM-types on Q are the
extensions to Qcm of a CM-type on some CM-subfield of N    Q.

C OROLLARY 1.31 The map .E; ˚ / 7! f  g defines a bijection from the set of isomor-
phism classes of primitive CM-pairs .E; ˚ / to the set of Gal. N
                                                              Q=Q/-orbits of CM-types on
  cm
Q .
                                                   N in the proposition.
P ROOF. Pass to the limit over all CM-subfields of Q                                                                2

E XAMPLE 1.32 From (1.12) we can read off the list of isomorphism classes of primitive
CM-pairs whose reflex field is contained in QŒe 2 i=13 .

R EMARK 1.33 Let .E; ˚ / be a CM-pair with reflex field contained in k, and let  be
the CM-type on k defined by an embedding W E ,! Q. N For any Q-algebra R and a ˝ r 2
        
.k ˝Q R/ ,
                                   def
                                       Y
                        N .a ˝ r/ D         al
                                                . a ˝ r/  . /
                                                        Wk!Q

is independent of , and equals N˚ .a ˝ r/.

R EMARK 1.34 As the above discussion makes clear, attached to a CM-pair .E; ˚ / there
is only an orbit of CM-types on the reflex field E  . However, when E is a subfield of Qal ,
there is a well-defined CM-type ˚  on E  corresponding to the given embedding of E,
called the reflex of ˚ (see 1.19).

E XERCISE 1.35 Rewrite this section for a k that is not necessarily Galois over Q.

Positive involutions and CM-algebras
Let B be an algebra (not necessarily commutative) over a field k. An involution of B is a
k-linear map b 7! b 0 W B ! B such that .ab/0 D b 0 a0 and .b 0 /0 D b for a; b 2 B. Because of
our convention on ring homomorphisms, 10 D 1 and so c 0 D c for c 2 k.
    Throughout this subsection, Q is a subfield of R. An involution 0 on a finite-dimensional
Q-algebra B is said to be positive if

                                               TrB=Q .b 0  b/ > 0                                              (13)
    9 For  2 Gal.Qal =Q/, E
                              D E and W E ! E is an isomorphism .E ; ˚ /.
   10 Consider, for example, a CM-pair .E; ˚/ and a fixed embedding of E into Qal . The composite


                                            .E; ˚/ 7!     7! .E ; ˚ /

sends .E; ˚ / to the reflex .E  ; ˚  / of its reflex .E  ; ˚  /. It is obvious from the definition of the reflex
CM-pair, that .E  ; ˚  / is a primitive CM-subpair of .E; ˚/, and therefore equals it if .E; ˚/ is primitive.
CHAPTER I. ANALYTIC THEORY                                                                     20


for every nonzero b 2 B. Note that

                           .b1 ; b2 / 7! TrB=Q .b10  b2 /W B  B ! Q

is Q-bilinear, and so it suffices to check (13) for the elements of a Q-basis for B. Therefore,
an involution on B is positive if and only if its linear extension to B ˝Q R is positive.

P ROPOSITION 1.36 Every finite-dimensional Q-algebra admitting a positive involution is
semisimple.

P ROOF. Let B admit a positive involution 0 , and let a be a nilpotent two-sided ideal in B.
                                                                             def
We have to show that a D 0. If not, there exists a nonzero a 2 a. Then b D a0 a 2 a and
is nonzero because TrB=Q .b/ > 0. As b D b 0 , TrB=Q .b 2 / > 0 and so b 2 ¤ 0; similarly
TrB=Q .b 4 / > 0 and so b 4 ¤ 0, etc., contradicting the nilpotence of a.                  2

P ROPOSITION 1.37 Let B be a finite-dimensional Q-algebra. The following conditions
on an involution 0 of B are equivalent:
  (a) B is semisimple and some faithful B-module V admits a positive definite symmetric
      Q-bilinear form . j /W V  V ! Q such that

                               .bujv/ D .ujb 0 v/, all b 2 B, u; v 2 V I                     (14)

  (b) every B-module admits a positive definite symmetric Q-bilinear form satisfying (14);
  (c) 0 is positive.

P ROOF. (a) H) (b). Every B-module is a direct summand of a direct sum of copies of V
(see p. 8), and the restriction of a bilinear form as in (a) to a B-submodule is of the same
type.
    (b) H) (c). We may suppose Q D R. Let W be a B-module. According to (b),
there exists a positive definite symmetric R-bilinear form . j /W W  W ! R satisfying (14).
Because Q D R, there exists an orthonormal basis e1 ; : : : ; en for W relative to . j /, and, for
any b 2 B, the trace of b 0 b on W is
                              X                   .14/ X
                                      .ei jb 0 bei / D         .bei jbei /,
                                  i                        i

which is > 0 unless b acts as zero on W . When we apply this remark with W D B, we find
that TrB=R .b 0 b/ > 0 unless b D 0.
    (c) H) (a). Proposition 1.36 shows that B is semisimple, and for V we can take B with
.ujv/ D TrB=Q .uv 0 /.                                                                  2

E XAMPLE 1.38 (a) For a totally real number field F , the identity involution is positive.
   (b) For a CM-field E, the involution E is positive.

P ROPOSITION 1.39 Every finite-dimensional commutative Q-algebra with positive invo-
lution is a product of pairs as in (1.38).

P ROOF. First consider an arbitrary finite-dimensional semisimple algebra B with involu-
tion 0 over a field k. The involution 0 permutes the set of simple two-sided ideals in B, from
which it follows easily that B decomposes (as a Q-algebra with involution) into a product
each of whose factors is either (a) a simple algebra with an involution or (b) the product of
two simple algebras interchanged by 0 .
CHAPTER I. ANALYTIC THEORY                                                                   21


    Next assume that B is commutative, that k D R, and that 0 is positive. Case (b) is
excluded11 , from which it follows that the only possibilities for the factors are .R; id/ or
.C; /.
    Finally assume that B is commutative, that k D Q, and that 0 is positive. Case (b) is
again excluded, and so we need consider only the case that B is a field. Then Aut.C/
acts transitively on the set of homomorphisms B ! C, and it follows that all factors of
.B;0 / ˝Q R are of the same type. If they are .R; id/, B is totally real and 0 D id; if they are
.C; /, B is a CM-field and 0 D .                                                             2

C OROLLARY 1.40 The CM-algebras are exactly the finite-dimensional commutative Q-
algebras admitting a (unique) positive involution that acts nontrivially on each factor.


2     Complex abelian varieties
Complex tori
A lattice  in a C-vector space V is the Z-submodule generated by an R-basis for V ,
i.e., such that R ˝Z  ' V . Equivalently, it is a discrete subgroup  of V such that V =
is compact (ANT 4.14). The quotient V = is a complex manifold with a distinguished
point, namely, the coset containing 0, and any pointed complex manifold isomorphic to
such a quotient is called a complex torus. Equivalently, a complex manifold M with a
distinguished point 0 is a complex torus if the exponential map
                                            expW Tgt0 .M / ! M
realizes M as the quotient of the complex vector space Tgt0 .M / by a lattice. In particular,
                                                                            '
we see that a complex torus has a canonical uniformization W V = ! M .
P ROPOSITION 2.1 Every complex torus is a compact connected complex Lie group. Con-
versely, if M is a compact connected complex Lie group, then the exponential map realizes
M as a complex torus.
P ROOF. The first assertion is obvious; for the second, see, for example, Mumford 1970, I,
1.                                                                                     2

P ROPOSITION 2.2 Let M ' V = and M 0 ' V 0 =0 be complex tori. A C-linear map
˛W V ! V 0 such that ˛./  0 defines a holomorphic map M ! M 0 sending 0 to 0, and
every holomorphic map M ! M 0 sending 0 to 0 is of this form (for a unique ˛).
                                                                        0
P ROOF. We choose bases, and identify V and V 0 with Cn and Cn respectively. The map
     ˛     0       0
Cn ! Cn ! Cn =0 is holomorphic, and it factors through Cn =. Because Cn = has
                                                              0
the quotient complex structure, the resulting map Cn = ! Cn =0 is holomorphic. Con-
                           0
versely, let 'W Cn = ! Cn =0 be a holomorphic map such that '.0/ D 0. Then Cn and
   0                                               0
Cn are universal covering spaces of Cn = and Cn =0 , and a standard result in topology
                                                                                       0
(Hatcher 2002, 1.33, 1.34) shows that ' lifts uniquely to a continuous map ˛W Cn ! Cn
such that ˛.0/ D 0:
                                                         ˛         0
                                            Cn                 Cn

                                                         '     0
                                          Cn =              Cn =0 :
    11 For if B D B  B , then .a; 0/.a; 0/0 D .0; 0/:
                   1   2
CHAPTER I. ANALYTIC THEORY                                                                                22


We have to show that ˛ is linear. Because the vertical arrows are local isomorphisms, ˛ is
holomorphic. For any ! 2 , the map z 7! ˛.z C !/ ˛.z/ is continuous and takes values
in 0  C; as Cn is connected and 0 is discrete, it must be constant. Therefore,
                                      @˛                @˛
                                          .z C !/           .z/ D 0;
                                      @zj               @zj
                     @˛                                                            0
and so, for each j , @z  defines a holomorphic function Cn = ! Cn , which must be con-
                       j
stant (because Cn = is compact). Write ˛ as an n0 -tuple .˛1 ; : : : ; ˛n0 / of holomorphic
                                                       @˛i
functions ˛i in n variables. Because ˛i .0/ D 0 and @z     is constant for each j , the power
                                          P              j        P
series expansion of ˛i at 0 is of the form j aij zj . Now ˛i and j aij zj are holomorphic
functions on Cn that coincide on a neighbourhood of 0, and so are equal on the whole of
Cn .                                                                                       2

C OROLLARY 2.3 Every holomorphic map M ! N of complex tori sending 0M to 0N is a
homomorphism. In particular, the group structure on M is uniquely determined by the zero
element.

    A Riemann pair .; J / is a free Z-module of finite rank  together with a complex
structure J on R ˝  (i.e., J is an R-linear endomorphism of  with square 1). A
homomorphism .; J / ! .0 ; J 0 / of Riemann pairs is a homomorphism ˛W  ! 0 of
Z-modules such that id ˝˛W R ˝  ! R ˝ 0 is C-linear:
                                                                 def
P ROPOSITION 2.4 The functor .; J /       M.; J / D .R ˝ ; J /= is an equivalence
from the category of Riemann pairs to the category of complex tori.

P ROOF. Proposition 2.2 says that the functor is fully faithful, and it is essentially surjective
by definition.                                                                                  2

E XAMPLE 2.5 Let .E; ˚ / be a CM-pair, and let  be a lattice in E, so that  ˝Z Q ' E.
                                                         .3/
Recall that ˚ defines an isomorphism E ˝Q R ! C˚ ,12 and so

                                                                         ˚
                             ˝Z R '  ˝Z Q ˝Q R ' E ˝Q R ' C˚ ,

from which  ˝Z R acquires a complex structure. Thus, from a CM-pair .E; ˚ / and a
lattice  in E, we get a Riemann pair .; J˚ /, and hence an abelian variety A˚ together
with a homomorphism i˚ W E ! End0 .A˚ / such that

                                           C˚            A˚
                                              z7!˚.a/z      i˚ .a/

                                           C˚            A˚

commutes for all a in
                                   fa 2 E j a  g ' End.A˚ /:
We also write ˚ for the map a 7! .'.a//' W  ! C˚ realizing  as a lattice in C˚ .
   12 Here C˚ is the set of maps ˚ ! C. In other words, it is a product of copies of C indexed by the elements

of ˚.
CHAPTER I. ANALYTIC THEORY                                                                            23


    An isogeny of complex tori is a surjective homomorphism with finite kernel. By an
“isogeny” we mean an invertible element of
                                                      def
                                  Hom0 .M; N / D Hom.M; N / ˝ Q:

Thus an “isogeny” M ! N need not be a map from M ! N , but some integer multiple
will be (in fact, an isogeny). More generally, by a “homomorphism” M ! N we mean an
element of Hom0 .M; N /.13

The cohomology of complex tori
By a real torus, we mean a quotient M D V = of a real vector space V by a lattice .
For example, the circle S 1 ' R=Z is a real torus. Then V is a universal covering space of
M with  as its group of covering transformations, and so 1 .M; 0/ '  (Hatcher 2002,
1.40). Therefore, (ib. 2A.1)
                                     H1 .M; Z/ '                                     (15)
and (Greenberg 1967, 23.14)

                                        H 1 .M; Z/ ' Hom.; Z/:                                     (16)

P ROPOSITION 2.6 For a real torus M ' V =, there is a canonical isomorphism

                            H n .M; Z/ ' Hom. n ; Z/;
                                              V

i.e., H n .M; Z/ is canonically isomorphic to the set of n-alternating forms        ! Z.

P ROOF. From (16), we see that
                          ^n                                ^n
                                        H 1 .M; Z/ '              Hom.; Z/.

Since14                          ^n
                                       Hom.; Z/ ' Hom. n ; Z/;
                                                       V

we see that it suffices to show that cup-product defines an isomorphism
                                ^n
                                     H 1 .M; Z/ ! H n .M; Z/.                                       (17)
   13 Equivalently, we could define a “homomorphism” M ! N to be a pair .a; m/ with a a homomorphism

M ! N and m an integer > 0, modulo the equivalence relation

                                       .a; m/  .b; n/ ” na D bm.

   14 For a free Z-module  of finite rank, the pairing

                                              Vn            Vn
                                                   _           !Z

determined by
                                 .f1 ^    ^ fn ; v1 ˝    ˝ vn / D det.fi .vj //
is perfect, because it is modulo p for every p — see Bourbaki 1958, 8. Here _ D Hom.; Z/ and “perfect”
means the discriminant is a unit in Z, so that the pairing defines an isomorphism
                                           Vn _
                                                ! Hom. n ; Z/:
                                                             V
CHAPTER I. ANALYTIC THEORY                                                                                        24


Let T be the class of topological manifolds M whose cohomology groups are free Z-
modules of finite rank and for which the maps (17) are isomorphisms for all n. Certainly,
the circle S 1 is in T (its cohomology groups are Z, Z, 0; : : :/, and the Künneth formula
(Hatcher 2002, 3.16 et seq.) shows that if M1 and M2 are in T , then so also is M1  M2 .
As a topological manifold, R2n =  .S 1 /2n , and so M is in T .                        2

R EMARK 2.7 Therefore,
                 ^n
                              HomR-linear .V; C/ ' H n .M; C/ ' HdR
                                                                 n
                                                                    .M /:
The composite isomorphism can be described as follows [to be added, cf. Debarre 1999].
    Every R-linear map V ! C can be written uniquely as the sum of a C-linear map and
a C-semilinear map (i.e., an additive homomorphism ˛W V ! C such that ˛.av/ D Nav for
a 2 C, v 2 V ). Thus,
                                  HomR .V; C/ D T ˚ TN ;
where
                                      T D HomC-linear .V; C/
                                      TN D HomC-semilinear .V; C/:
Therefore,               ^n                       M                 ^p        ^q
                              HomR .V; C/ '                              T˝         TN :
                                                       pCqDn
On the other hand, there is the Hodge decomposition
                          M
               n
             HdR .M / '             H p;q .M /; H p;q .M / D H q .M; ˝ p /:
                                   pCqDn

P ROPOSITION 2.8 The two decompositions correspond under the isomorphism in (2.7); in
particular,              ^p      ^q
                             T˝       TN ' H q .M; ˝ p /.
    We shall describe the Hodge structure later. Here we only need that
                                      H 1 .M; R/ '               1
                                                            .M; ˝hol /,
so that
                             H 1 .M; C/ '              1
                                                  .M; ˝hol          1
                                                           / ˚ .M; ˝hol /;
and, dually,
                                  H1 .M; C/ ' Tgt0 .A/ ˚ Tgt0 .A/.                                           (18)

Hermitian forms and alternating forms
To give a complex vector space amounts to giving a real vector space V together with an
endomorphism J W V ! V such that J 2 D 1. Ap   hermitian form on .V; J / is an R-bilinear
mapping . j / W V  V ! C such that .J ujv/ D     1.ujv/ and .vju/ D .ujv/. When we
write15                            p
                   .ujv/ D '.u; v/    1 .u; v/; '.u; v/, .u; v/ 2 R,                 (19)
  15 For example, let V D C, so .zjz 0 / D az zN 0 for some a > 0. Then the decomposition (19) is (i D
                                                                                                         p
                                                                                                             1)
                                         0    0         0       0         0     0
                           a.x C iy/.x       iy / D a.xx C yy / i a.xy yx / :
                           „       ƒ‚          … „ ƒ‚ … „ ƒ‚ …
CHAPTER I. ANALYTIC THEORY                                                                                 25


the pairings u; v 7! '.u; v/ and u; v 7!            .u; v/ are R-bilinear, and
                 ' is symmetric                                  '.J u; J v/ D '.u; v/;                  (20)
                     is alternating                                .J u; J v/ D      .u; v/;             (21)
                    .u; v/ D '.u; J v/;                              '.u; v/ D       .u; J v/:           (22)
16 As .uju/ D '.u; u/, . j / is positive definite if and only if ' is positive definite. Conversely,

if ' satisfies (20) (resp.       satisfies (21)), then the formulas (22) and (19) define a hermitian
form:
                             p                                                     p
      .ujv/ D '.u; v/ C           1'.u; J v/       (resp. .ujv/ D      .u; J v/         1 .u; v/).       (23)

Riemann forms
Let .; J / be a Riemann pair. An integral Riemann form (or just Riemann form) for
.; J / is an alternating Z-bilinear form W    ! Z such that
                                  .x; y/ 7!       R .x; Jy/W R  R ! R

 is symmetric and positive definite. Equivalently (see above), it is the imaginary part of a
positive definite hermitian form that takes integer values on .
     Let be an alternating Z-bilinear form on , and let J .x; y/ D R .x; Jy/. Then J
is symmetric and positive definite if and only if
   ˘    R .J x; Jy/ D R .x; y/ for all x; y 2 R , and
  ˘      R .x; J x/ > 0 for all nonzero x 2 R .
    A Riemann form is nondegenerate, and so, for any ˛ 2 End0 .; J /, there exists a
unique ˛ 0 2 End0 .; J / such that
                                  .˛x; y/ D        .x; ˛ 0 y/;   all x; y 2 Q :
The map ˛ 7! ˛ 0 is an involution on End0 .; J /, called the Rosati involution (relative to
 ). For x; y 2 R and ˛ 2 End0 .; J /,

               J .˛x; y/ D       .J˛x; y/ D         .˛J x; y/ D     .J x; ˛ 0 y/ D            0
                                                                                       J .x; ˛ y/;

and so the Rosati involution is positive (1.37).
   By a rational Riemann form, we mean an alternating Q-bilinear form W Q  Q !
Q such that .x; y/ 7! .x; Jy/W R  R ! R is symmetric and positive definite. Then
            1
  .; /  m  Z for some positive integer m, and m is an (integral) Riemann form.
E XAMPLE 2.9 Let .E; ˚ / be a CM-pair, and write aN for E a. Let  be a lattice in E, and
let .; J˚ / be the corresponding Riemann pair (2.5). Then
                                            def
                                         R D fa 2 E j a  g
is an order in OE , and R  End.; J˚ /. Therefore, E  End0 .; J˚ /. We wish to de-
termine the rational Riemann forms on .; J˚ / for which the Rosati involution stabilizes
E (and therefore acts on it as E — recall (1.40) that E is the only positive involution on
E). To give such a form amounts to giving a nondegenerate Q-bilinear form W E  E ! Q
such that
   16 Should re-think these signs. Perhaps remove the   from (19) and define J to be .J x; y/ D        .x; Jy/.
Then need to choose ˛ in (2.9) so that =.'˛/ < 0, and need to change (Deligne’s) definition of the polarization
of a Hodge structure (so that .2 i/n .h.i/x; y/ > 0 rather than .2 i/n .x; h.i/y/ > 0; Deligne 1979).
CHAPTER I. ANALYTIC THEORY                                                                 26


  (a)    .ax; y/ D    .x; Nay/, all a; x; y 2 E,
  (b)    .x; y/ D     .y; x/, all x; y 2 E,
  (c)    .J˚ x; J˚ y/ D    .x; y/, all x; y 2 E ˝ R,
 (d) .x; J˚ x/ > 0 for all nonzero x 2 E ˝ R.
The following statements are left as an easy exercise for the reader (see the appendix).
  ˘ For any ˛ 2 E  ,
                              .x; y/ 7! TrE=Q .˛x Ny/W E  E ! Q                         (24)
        is a nondegenerate Q-bilinear form satisfying (a), and every such form arises in this
        way from a unique ˛.
  ˘ Condition (b) holds for the form (24) if and only if ˛N D ˛.
  ˘ Condition (c) holds automatically for the form (24).
   ˘ Condition (d) holds for the form (24) if and only if =.'.˛// > 0 for all ' 2 ˚.
We conclude that the rational Riemann forms for .; J˚ / are in one-to-one correspondence
with the ˛ 2 E  such that ˛N D ˛ and =.'.˛// > 0 for all ' 2 ˚.
     Let F be the product of the largest totally real subfields of the factors of E. Then (cf.
1.4) E D F Œ˛ with ˛ 2 2 F , which implies that ˛N D ˛. The weak approximation theorem
(ANT, 7.20) shows that ˛ can chosen so that =.'.˛// > 0 for all ' 2 ˚. Thus, elements ˛
with the required properties certainly exist, and so .; J˚ / is polarizable.
     Let ˛ be one element of E  such that ˛N D ˛ and =.'.˛// > 0 for all ' 2 ˚. Then the
other such elements are exactly those of the form ˛a with a a totally positive element of F
(i.e., aN D a and '.a/ > 0 for all 'W F ! R).

Abelian varieties
D EFINITION 2.10 An abelian variety is a complex torus that admits a Riemann form.

E XAMPLE 2.11 For any CM-pair .E; ˚ / and lattice  in E, the complex torus C˚ =˚./
admits a polarization (2.9), and so is an abelian variety.
                                                          def
   If A ' V = is an abelian variety, then so also is A_ D V 0 =0 , where V 0 is the space of
semilinear maps V ! C and 0 D ff 2 V _ j f ./  Zg. Moreover, an integral Riemann
form on A defines an isogeny

                              Œa 7! .Œx 7! Œ .a; x//W A ! A_ :

Here Œa D a C  2 A.

T HEOREM 2.12 (P OINCAR É REDUCIBILITY THEOREM ) For any abelian subvariety B of
an abelian variety A, there exists an abelian variety B 0  A such that B \ B 0 is finite and
B C B 0 D A, i.e., such that .b; b 0 / 7! b C b 0 W B  B 0 ! A is an isogeny.

P ROOF. Let A ' V = be the canonical uniformization of A, and let W  V be the tangent
space at 0 of B  A; then

                               B D W = . \ W /  V = D A:
CHAPTER I. ANALYTIC THEORY                                                                                27


Choose an integral Riemann form for A, and let W ? be the orthogonal complement to
W under R . Then W ? is stable under J , and  \ W ? is a lattice in W ? because it has
rank
                         rank  rank  \ W D 2 dimC W ? .
                                                                         def
As j \ W ? is a Riemann form for . \ W ? ; J jW ? /, B 0 D W ? =W ? \  is an abelian
subvariety of A. Moreover, B \ B 0 is finite.                                         2

     An abelian variety A is said to be simple if it has no proper nonzero abelian sub-
varieties. It follows    easily from the theorem that every abelian variety A is isogenous
to a product Ari i of powers of nonisogenous simple abelian varieties Ai ; the ri are
                Q
uniquely determined and the Ai are uniquely determined up to isogeny. Moreover, each
End0 .Ai / isQa division algebra, End0 .Ari i / is equal to the matrix algebra Mri .Ai /, and
End0 .A/ D End0 .Ari i /. In particular, End0 .A/ is semisimple.17
                                                                                      p
E XERCISE 2.13 Let A be the quotient of C2 by the lattice generated by .i; 0/, . 2; i /,
.1; 0/, .0; 1/, and let B be the quotient of C by the lattice generated by i and 1. Show that
the image BN of the map
                                      z 7! .z; 0/W B ! A
is a complex subtorus of the complex torus A for which there does not exist a complex
subtorus B 0  A such that BN \ B 0 is finite and BN C B 0 D A. (Hence, no Riemann form
exists for A. In fact, most complex tori are not abelian varieties.)


3     Abelian varieties with complex multiplication
Definition of CM abelian varieties
P ROPOSITION 3.1 For any abelian variety A,

                                       2 dim A  ŒEnd0 .A/W Qred :

When equality holds, End0 .A/ is a product of matrix algebras over fields.18
P ROOF. As End0 .A/ is a semisimple Q-algebra acting faithfully on the 2 dim A-dimensional
Q-vector space H1 .A; Q/, this follows from (1.2).                                      2

D EFINITION 3.2 A complex abelian variety A is said to have complex multiplication (or
be of CM-type, or be a CM abelian variety) if

                                       2 dim A D ŒEnd0 .A/W Qred .

P ROPOSITION 3.3 The following conditions on an abelian variety A are equivalent:
  (a) A has complex multiplication;
    (b) End0 .A/ contains an étale subalgebra of degree 2 dim A over Q;
    (c) for any Weil cohomology X       H  .X / with coefficient field ˝, the centralizer of
        End0 .A/ in End˝ .H 1 .A// is commutative (and equals C.A/ ˝Q ˝, where C.A/ is
        the centre of End0 .A/).
    17 This also follows directly from the fact that the Rosati involution on End0 .A/ defined by any Riemann

form is positive.
   18 Recall (see Notation) that fields are commutative.
CHAPTER I. ANALYTIC THEORY                                                                    28


P ROOF. (a) ” (b). According to (1.3), the degree of a maximal étale subalgebra is
ŒEnd0 .A/W Qred .
    (a) ” (c). From the definition of a Weil cohomology, one deduces that H 1 .A/ has
dimension 2 dim A over ˝, and that End0 .A/ ˝Q ˝ acts faithfully on it. Thus, if (a) holds,
then End0 .A/ ˝Q ˝ is a product of matrix algebras over fields and H 1 .A/ is reduced (1.2).
From this, (c) follows. The converse is equally easy.                                     2

E XAMPLE 3.4 For any CM-pair .E; ˚ / and lattice  in E, the abelian variety A˚ D
C˚ =˚./ (see 2.11) has complex multiplication because End0 .A/ contains the étale sub-
algebra E, which has degree 2 dim A˚ over Q.
R EMARK 3.5 Let A  i Ani i be the decomposition of A (up to isogeny) into a product
                         Q

Q isotypic abelian varieties. Then Di D
of                                           End0 .Ai / is a division algebra, and End0 .A/ '
                                          0
   i Mni .Di / is the decomposition End .A/ into a product of simple Q-algebras. From
(3.3), we see that A has complex multiplication if and only if Di is a commutative field of
degree 2 dim Ai for all i . In particular, a simple abelian variety A has complex multipli-
cation if and only if End0 .A/ is a field of degree 2 dim A over Q, and an arbitrary abelian
variety has complex multiplication if and only if each simple isogeny factor does.
P ROPOSITION 3.6 (a) A simple abelian variety A has complex multiplication if and only
if End0 .A/ is a CM-field of degree 2 dim A over Q.
    (b) An isotypic abelian variety A has complex multiplication if and only if End0 .A/
contains a field of degree 2 dim A over Q (which can be chosen to be a CM-field invariant
under some Rosati involution).
    (c) An abelian variety A has complex multiplication if and only if End0 .A/ contains
an étale Q-algebra (which can be chosen to be a CM-algebra invariant under some Rosati
involution) of degree 2 dim A over Q (in which case H1 .A; Q/ is free of rank 1 over the
algebra).
P ROOF. (a) After the remark, it remains to show that if End0 .A/ is a field of degree 2 dim A
then it is CM. We know that it is either totally real or CM because it is stable under the Rosati
involutions (1.39), and Lemma 3.7 below shows that it must be the former.
                                                             0
    (b) Write A  Am  0 with A0 simple. Then E0 D End .A0 / is a CM-field. Let F be a
                                                                                   def
totally real field of degree m over Q that it linearly disjoint from E0 . Then E D E0  F is a
CM-field of degree 2 dim A, and the choice of an E0 -basis for it defines an embedding of it
into Mm .E0 / ' End0 .A/. Moreover, (2.9) provides A with a polarization under which E
is stable.
     (c) Follows from (b) (H1 .A; Q/ is free of rank 1 because E acts faithfully on it).    2

L EMMA 3.7 Let F be a subfield of End0 .A/, some abelian variety A. If F has a real prime,
then ŒF W Q divides dim A.
P ROOF. For any endomorphism ˛ of A, there is a (unique) polynomial P˛ .T / 2 QŒT  of
degree 2 dim A such that, for all rational numbers r, P˛ .r/ D deg.˛ rA /; moreover, P˛ .T /
is the characteristic polynomial of ˛ on H1 .A; Q/ (see Milne 1986, Section 12, for a proof
in a more abstract setting).
                                                            def
     Note that H1 .A; Q/ is a vector space of dimension m D 2 dim A=ŒF W Q over F , and so,
for any ˛ 2 End.A/ \ F , P˛ .T / is the mth -power of the characteristic polynomial of ˛ in
F=Q. In particular,
                                  NmF =Q .˛/m D deg.˛/  0:
CHAPTER I. ANALYTIC THEORY                                                                    29


However, if F has a real prime, then ˛ can be chosen to be large and negative at that prime
and close to 1 at the remaining primes (weak approximation theorem, ANT 7.20), so that
NmF =Q .˛/ < 0. This gives a contradiction unless m is even.                              2

R EMARK 3.8 A subalgebra E of End0 .A/ such that H1 .A; Q/ is a free E-module of rank
1 need not be CM, even when it is a field. Consider, for example, an elliptic curve A0 with
complex multiplication by an imaginary quadratic field E0 , and let A D Am 0 . Let F be any
field of degree m over Q and linearly disjoint from E0 , and embed F into GLm .Q/, hence
into GLm .E0 /. Then H1 .A; Q/ is of dimension 1 over the field E D E0 F  End0 .A/, but
E is CM if and only if F is totally real or CM (note that a field with Galois group A5 and
can not be CM).

R EMARK 3.9 An abelian variety A with complex multiplication by E is isogenous to a
principal abelian variety, i.e., an abelian variety on which the full ring of integers of E acts.
To see this, write A D C˚ =˚.a/ with a a lattice in E, and consider C˚ =˚.b/, where b is
an ideal contained in a.

E XERCISE 3.10 Let L be a simple Q-algebra of finite degree d 2 over its centre F , and let
A be an abelian variety containing L in its endomorphism algebra.
  (a) Show that for any semisimple commutative Q-subalgebra R of End0L .A/, dimQ R 
      .2 dim A/=d , and that equality holds for some R if and only A has complex multipli-
      cation.
  (b) Let 0 be a Rosati involution on End0 .A/ stabilizing L; show that, if A has complex
      multiplication, then there is an R as in (a) that is stabilized by 0 .

The reflex field of an abelian variety with complex multiplication
Let E0 be the centre of End0 .A/. There exists a CM-type ˚0 on E0 with the following
property: suppose A is of CM-type .E; ˚ / (relative to E ,! End0 .A/); then .E; ˚ / extends
.E0 ; ˚0 /. Therefore, the reflex field of .E0 ; ˚0 / equals the reflex field of any such .E; ˚ /
(1.18c). We call it the reflex field of A.

Classification up to isogeny
 3.11 Let A be an abelian variety with complex multiplication, so that End0 .A/ contains
a CM-algebra E for which H1 .A; Q/ is free E-module of rank 1, and let ˚ be the set of
homomorphisms
L               E ! C occurring in the representation of E on Tgt0 .A/, i.e., Tgt0 .A/ '
   '2˚ C' , where C' is a one-dimensional C-vector space on which a 2 E acts as '.a/.
Then, because
                            H1 .A; R/ ' Tgt0 .A/ ˚ Tgt0 .A/                          (25)
(see (25)) ˚A is a CM-type on E, and we say that, A together with the injective homomor-
phism i W E ! End0 .A/, is of CM-type .E; ˚ /.
    Let e be a basis vector for H1 .A; Q/ as an E-module, and let a be the lattice in E such
that ae D H1 .A; Z/. Under the isomorphism (cf. (25))
                                       M             M
                          H1 .A; R/ '           C' ˚         C' ;
                                               '2˚                    '2˚

                             e˝1     ! .: : : ; e' ; : : : I : : : ; eı' ; : : :/
CHAPTER I. ANALYTIC THEORY                                                                                30


where each e' is a C-basis for C' . The e' determine an isomorphism
                                         M
                              Tgt0 .A/ '         C' ' C˚ ;
                                                       '2˚

and hence a commutative square of isomorphisms in which the top arrow is the canonical
parametrization:
                               Tgt0 .A/=         A
                                                                                                        (26)

                                         C˚ =˚.a/               A˚ :

P ROPOSITION 3.12 The map .A; i / 7! .E; ˚ / gives a bijection from the set of isogeny
classes of pairs .A; i/ to the set of isomorphism classes of CM-pairs, with inverse .E; ˚ / 7!
.A˚ ; i˚ /.

P ROOF. We have well-defined maps between the two sets, whose composites we shall
show to identity maps. Let .E; ˚ / be the CM-type of .A; i /; then (26) shows that .A; i / 
.A˚ ; i˚ /. In the other direction, it is obvious that .A˚ ; i˚ / is of CM-type .E; ˚ /.   2

   We make this classification more precise in the case of simple abelian varieties with
complex multiplication.

P ROPOSITION 3.13 Let A be a simple abelian variety with complex multiplication, and
let E D End0 .A/. Then .E; ˚A / is a primitive CM-type, and the map A 7! .E; ˚A / de-
fines a bijection from the set of isogeny classes of simple abelian varieties with complex
multiplication to the set of isomorphism classes of primitive CM-pairs.

P ROOF. Because A is simple, E is a field. If .E; ˚A / is not primitive, and so is the exten-
sion of a CM-type .E0 ; ˚0 / with E0 a proper subfield of E, then A will be isogenous to
  ŒE WE 
A˚0 0 , and so is not simple.                                                              2

C OROLLARY 3.14 The simple abelian varieties with complex multiplication are classi-
fied up to isogeny19 by the -orbits of CM-types on Qcm , where D Gal.Qcm =Q/ (or
Gal.Qal =Q/).

P ROOF. Combine Proposition 3.13 with Proposition 1.30.                                                     2

C OROLLARY 3.15 The pairs .A; / consisting of a simple CM abelian variety and an em-
bedding End0 .A/ ,! Qal are classified up to isogeny by the CM-types on Qcm .

R EMARK 3.16 Let A be a simple abelian variety corresponding, as in the Corollary, to the
  -orbit , and let E D End0 .A/. For each 2 , let E be the fixed field of f 2 j
 D g. Then, as runs through , E runs through the conjugates of E in Qal .
    19 We mean by this that there is a canonical map sending a simple abelian variety with complex multiplica-

tion to an orbit of CM-types whose fibres are exactly the isogeny classes.
CHAPTER I. ANALYTIC THEORY                                                                    31


Classification up to isomorphism
Let .A; i/ be of CM-type .E; ˚ /. Let e be an E-basis element of H1 .A; Q/, and set
H1 .A; Z/ D ae with a a lattice in E. We saw in (3.11) that e determines an isomorphism
                                                         def
                          W .A˚ ; i˚ / ! .A; i /;   A˚ D C˚ =˚.a/:

Conversely, every isomorphism C˚ =˚.a/ ! A commuting with the actions of E arises in
this way from an E-basis element of H1 .A; Q/, because
                                                     
                                E ' H1 .A˚ ; Q/ ' H1 .A; Q/:

If e is replaced by ae, a 2 E  , then  is replaced by  ı a 1 .
     We use this observation to classify triples .A; i; /, where A is an abelian variety,
i W E ! End0 .A/ is a homomorphism making H1 .A; Q/ into a free module of rank 1 over
the CM-algebra E, and is a rational Riemann form whose Rosati involution stabilizes
i.E/ and induces E on it.
     Let W C˚ =˚.a/ ! A be the isomorphism defined by some basis element e of H1 .A; Q/.
According to (2.9), there exists a unique element t 2 E  such that .xe; ye/ D TrE=Q .tx N    y/.
The triple .A; i; / is said to be of type .E; ˚ I a; t / relative to  (cf. Shimura 1971, Section
5.5 B).

P ROPOSITION 3.17 The type .E; ˚ I a; t / determines .A; i; / up to isomorphism. Con-
versely, .A; i; / determines the type up to a change of the following form: if  is replaced
by  ı a 1 , a 2 E  , then the type becomes .E; ˚ I aa; t =a Na/. The quadruples .E; ˚ I a; t /
that arise as the type of some triple are exactly those in which .E; ˚ / is a CM-pair, a is a
lattice in E, and t is an element of E  such that E t D t and =.'.t // > 0 for all ' 2 ˚.

P ROOF. Routine verification.                                                                  2



4    Mumford-Tate groups
Review of algebraic groups of multiplicative type
For algebraic groups, we use the terminology from iAG. In particular, by an algebraic group
over a field k, we mean a scheme G of finite type over k together with a morphism

                                        mW G  G ! G

such that, for all k-algebras R, .G.R/; m.R// is a group. An algebraic group over k defines
a functor from k-algebras to groups, and such a functor arises from an affine algebraic group
if it is represented by a finitely generated k-algebra. The multiplication map m on an affine
                                                                                           def
algebraic group G over k defines a comultiplication map W kŒG ! kŒG˝kŒG on kŒG D
OG .G/, which makes it into a Hopf algebra, and every finitely generated commutative Hopf
algebra over k arises from an affine algebraic group over k. An algebraic group over a field
of characteristic zero is automatically smooth. See iAG, 1.1, 1.4, 3.6, 3.23.
      Let M be a finitely generated abelian group M (written multiplicatively). The functor
D.M / of k-algebras

                R    Hom.M; R /          (homomorphisms of abelian groups)
CHAPTER I. ANALYTIC THEORY                                                                                32


is represented by the k-algebra kŒM , which, as a k-vector space, has basis the elements of
M , and which has the multiplication
                           X           X          X
                               ai mi    bj nj D      ai bj mi nj :

Therefore D.M / is an algebraic group. The affine algebraic groups G arising in this way
are called diagonalizable groups. The Hopf algebra structure on kŒM  is
                                                                                1
                            .m/ D m ˝ m;         .T / D 1;      S.T / D T         :

The diagonalizable groups are exactly those whose Hopf algebra is generated by “group-
like” elements, i.e., elements m such that .m/ D m ˝ m. The group-like elements in kŒM 
are exactly the elements of M . See iAG, 12b,c.
    The diagonalizable group defined by .Z; C/ is Gm , which represents the functor of
k-algebras
                                          R R :
Its Hopf algebra is kŒT; T      1  with

                                                                                1
                            .T / D T ˝ T;        .T / D 1;     S.T / D T          :

For any affine algebraic group G, there is an isomorphism

                   7! T ı W Hom.G; Gm / ! fgroup-like elements in kŒGg:

    Let G be an affine algebraic group over k, and let m be a group-like element in kŒG.
As G.R/ D Homk-alg .kŒG; R/, the element m defines a map g 7! g.m/W G.R/ ! R for
every k-algebra R. For any k-vector space V , the maps

                             .g; v/ 7! g.m/  vW G.R/  V .R/ ! V .R/

define a representation of G on V — we say that G acts on V through m. Let W G !
GLV be a representation of G on a finite-dimensional k-vector space V . For each group-
like element m 2 kŒG, there is a largest subspace Vm of V on which G acts through m,
and GLis diagonalizable if and only if every representation decomposes into a direct sum
V D m2M Vm (iAG 12.12).20 Thus, to give a representation of a diagonalizable group
G D D.M / on a vector space amounts to giving an M -gradation of the vector space.
    For simplicity, we now assume k to have characteristic zero.
    Fix an algebraic closure k al of k and let D Gal.k al =k/. An affine algebraic group G
over k is said to be of multiplicative type if Gk al is diagonalizable. The functor

                                   G       X  .G/ D Hom.Gk al ; Gm /

defines a contravariant equivalence from the category of algebraic groups of multiplicative
type over k to the category of finitely generated abelian groups with a continuous action
of . Because X  .G/ is finitely generated, “continuous” simply means that some open
subgroup of acts trivially. To give a representation of a group of multiplicative type G
on a k-vector space V is the same as giving a X  .G/-gradation
                                            M
                                V ˝k k al D         
                                                        V
                                                         2X .G/
   20 To say that V D
                        L
                         m Vm means that there is a basis for V for which .G/  D (the group of invertible
diagonal matrices). Therefore, G is diagonalizable if and only if every representation of G is diagonalizable.
CHAPTER I. ANALYTIC THEORY                                                                       33


on V ˝k k al such that      permutes the subspaces V according to the rule,

                                                  V D V

(iAG, 12.23).
    A torus over k is an algebraic group T that becomes isomorphic to a product of copies
of Gm over k al . In other words, it is a group of multiplicative type G such that X  .G/ is
torsion free. A torus T is said to be split if acts trivially on X  .T /. In characteristic zero,
the tori are the connected algebraic groups of multiplicative type.
    For a finite field extension K=k, we let .Gm /K=k denote the torus over k obtained by
restriction of scalars from Gm over K (iAG, 2i). It represents the functor R .K ˝ R/ ,
and
                                                  al
                     X  ..Gm /K=k / D ZHomk .K;k /        (as -modules).
In other words, a character of .Gm /K=k is a finite sum
             X
                        al
                           n. /; n. / 2 Z;  a k-algebra homomorphisms,
                    WK!k

which acts as                    Y
                      c ˝ r 7!      .c/n. /  rW .K ˝ R/ ! .k al ˝ R/ .
                                     
Let V be a vector space over k. A decomposition k al ˝k V D WK!k al V such that
                                                                      L
 V D V  for all  2 Gal.k al =k/ defines a representation of .Gm /K=k on V .
    Let k al be an algebraic closure of k, and let D Gal.k al =k/. For any affine algebraic
group G, the pairing
                                   def
                  .; / 7! h; i D  ı W X  .G/  X .G/ ! End.Gm / ' Z

is bi-additive and    -equivariant, i.e.,

            h; i D h; i;            2 X  .G/;           2 X .G/;    2 Gal.k al =k/:

For groups G of multplicative type, it is non-degenerate in the sense that

           7! h; iW X .G/ ! Hom.X  .G/; Z/                     .Z-module homomorphisms)

is an isomorphism of ZŒ -modules.
    Let W G ! GLV be a representation of an algebraic group G. Then  applied to the
“universal element”
                          idkŒG 2 Endk-alg .kŒG/ D G.kŒG/
is a linear map
                                                       .idkŒG /
                                 kŒG ˝k V                 ! kŒG ˝k V
whose restriction to V ,
                                           W V ! kŒG ˝k V
                                           Q
determines  (iAG, 4.1). An endomorphism ˛ of the k-vector space V is an endomorphism
of the representation .V; / if and only if the diagram
                                                  Q
                                          V                kŒG ˝k V
                                              ˛                     id ˝˛
                                                  Q
                                          V                kŒG ˝k V
CHAPTER I. ANALYTIC THEORY                                                                  34


commutes. This is a linear condition on ˛, and so, for any field K containing k,
                                End.VK ; K / D End.V; / ˝k K:                           (27)
P ROPOSITION 4.1 Let G be a group of multiplicative type over a field k of characteristic
zero. For any representation W G ! GLV ,
                               ŒEnd.V; /W kred D dim V:
                                           L
P ROOF. If G is diagonalizable, then V D m Vm (sum over the group-like elements of
kŒG), and                   Y                         Y
                End.V; / '        Endk-linear .Vm /     Mdim.Vm / .k/;
                                     m                            m
from which the statement follows. In the general case, G becomes diagonalizable over a
finite extension K of k, and
                                                                      .1/
         dimk V D dimK .V ˝k K/ D ŒEnd.V; / ˝k KW Kred D ŒEnd.V; /W kred .               2



CM-pairs and tori
The Deligne torus S is defined to be .Gm /C=R . Thus S is a torus over R such that S.R/ D
C . There is a canonical isomorphism SC ' Gm  Gm , defined by characters z and zN of S,
such that the map S.R/ ! S.C/ induced by R ,! C is a 7! .a; Na/. Clearly, z and zN form a
basis for X  .S/. Let W Gm ! SC ' Gm  Gm be the cocharacter a 7! .a; 1/ of S, so
                               .z ı /.a/ D a;        .zN ı /.a/ D 1:
                                 E
                      Q and let T D .Gm /E=Q . As noted
   Let .E; ˚/ be a CM-pair,                             in 1 (3), ˚ defines an
isomorphism E ˝Q R ! '2˚ C, and hence an isomorphism21

                                                TRE ' S˚ :                                (28)
Define
                                               h˚ W S ! TRE
to be the homomorphism whose composite with (28) is
                                           z 7! .z; : : : ; z/:
                                Q
The isomorphism E ˝Q C '            '2I C, I D Hom.E; C/, defines an isomorphism

                                           TCE ' .Gm /I :
The cocharacter
                                         def
                                    ˚ D h˚ ı W GmC ! TCE
corresponding to h˚ satisfies
                                                 
                                                     z if ' 2 ˚
                                  ˚ .z/' D
                                                     1 if ' … ˚:
    Recall that the reflex field E  of .E; ˚ / is the subfield of C generated by the elements
                                    X
                                             '.a/; a 2 E:
                                           '2˚

It can also be described as the field of definition of ˚ :
  21 By S˚ we mean a product of copies of S indexed by ˚.
CHAPTER I. ANALYTIC THEORY                                                               35


The reflex norm in terms of tori
                                      
Let .E; ˚/ be a CM-pair, and let T E D .Gm /E  =Q and T E D .Gm /E=Q . The composite
of the homomorphisms
                                          ˚     NmE  =Q
                                         E
                               Gm=E  ! T=E        ! TE
is the reflex norm N˚ . Thus, for a 2 E  ,

                              N˚ .a/ D NmE  ˝Q E=E ˚ .a/.

Complex multiplication in terms of tori
Let A be an abelian variety, and let V = ' A be its canonical parametrization. Then

                                        ' H1 .A; Z/
                                    ˝ R ' H1 .A; R/
                                    ˝ R ' V ' Tgt0 .A/

and so
                                   H1 .A; R/ ' Tgt0 .A/:
In particular, H1 .A; R/ acquires the structure of a complex vector space from its identi-
fication with Tgt0 .A/. Also End.A/ (resp. End0 .A/) consists of the endomorphisms of
 ' H1 .A; Z/ (resp. H1 .A; Q/) whose linear extensions to H1 .A; R/ ' Tgt0 .A/ are C-
linear.

P ROPOSITION 4.2 An abelian variety A has complex multiplication if and only if there
exists a torus T  GLH1 .A;Q/ such that T .R/ contains all homotheties v 7! zvW H1 .A; R/ !
H1 .A; R/, z 2 C .

P ROOF. If A has complex multiplication, then there exists an étale Q-algebra E  End0 .A/
for which H1 .A; Q/ is a free E-module of rank 1. The action of E ˝ R on H1 .A; R/ com-
mutes with that of C, and so C is contained in the centralizer of E ˝ R in EndR .H1 .A; R//,
which is E ˝ R itself. Therefore we can take T D .Gm /E=Q  GLH1 .A;Q/ .
    Conversely, let W T ,! GLH1 .A;Q/ be a subtorus. If C  T .R/, then End0 .A/ 
End.H1 .A; Q/; /, and Proposition 4.1 shows that A has complex multiplication.           2


Mumford-Tate groups
By a rational Riemann pair .V; J /, we mean a finite-dimensional Q-vector space V to-
gether with a complex structure J on V ˝ R. By a Riemann form on .V; J /, we mean an
alternating bilinear form on V such that
   ˘     .J x; Jy/ D .x; y/ for all x; y 2 VR , and
  ˘      .x; Jy/ > 0 for all nonzero x in VR .
                def
Then J .x; y/ D R .x; Jy/ is a symmetric positive-definite form on VR .
    Let S D .Gm /C=R , so that S.R/ D C . There is a homomorphism hW S ! GLVR such
that h.z/ acts on VR as multiplication by z.

L EMMA 4.3 Let .V; J / be a rational Riemann pair. The following conditions on an alge-
braic subgroup G of GLV are equivalent:
CHAPTER I. ANALYTIC THEORY                                                                              36


  (a) H.R/ contains the homotheties v 7! zv; z 2 C ;
  (b) H.Q/ contains the homotheties v 7! zv, z 2 Q , and H.R/ contains J ;
  (c) H contains h.S/.

P ROOF. Certainly (a) implies (b). If H.Q/ contains the homotheties v 7! zv, z 2 Q , then
H  Gm . As h.S/ is generated by Gm and J , (b) implies (c). Finally, (c) implies (a)
because h.S/.R/ D C .                                                                   2

D EFINITION 4.4 The Mumford-Tate group of a rational Riemann pair .V; J / is the small-
est algebraic subgroup of GLV satisfying the equivalent conditions of Lemma 4.3. The
Mumford-Tate group of a complex abelian variety is the Mumford-Tate group of the asso-
ciated rational Riemann pair .H1 .A; Q/; J /.

    Thus, MT.A/ is the smallest algebraic subgroup G of GLH1 .A;Q/ such that G.R/ con-
tains all homotheties

                           v 7! zvW H1 .A; R/ ! H1 .A; R/;           z 2 C :                         (29)

   If G1 and G2 are algebraic subgroups of GLV satisfying the conditions of (4.3), then
G1 \ G2 has the same property, and so there is certainly is a smallest subgroup with this
property.

P ROPOSITION 4.5 A Q-subspace W of V is stable under G D MT.V; J / if and only if
W ˝ R is a C-subspace of V ˝ R.

P ROOF. Recall (iAG, 4.3), that there exists an algebraic subgroup H of GLV , called the
stabilizer of W , such that

              H.R/ D f˛ 2 GL.V ˝ R/ j ˛.W ˝ R/ D W g;                   all Q-algebras R:             (30)

    Let W be a Q-subspace of V and let H be the stabilizer of W in GLV . Then

W ˝ R is a C-subspace of H1 .A; R/ ” H.R/ contains the homotheties v 7! zv, z 2 C;
                                            ” H  MT.V; J /;
                                            ” W is stable under MT.V; J /:                               2


P ROPOSITION 4.6 The Mumford-Tate group of a Riemann pair is connected, and it is re-
ductive if the riemann pair is polarizable.

P ROOF. Because h.S/ is connected, if HR  h.S/, then .HR /ı  h.S/, and obviously
(H ı /R  .HR /ı (in fact, the two are equal iAG 1.34).
    Let H D MT.V; J /. Every subspace W of V stable under H has a complement sta-
ble under H , namely, its orthogonal complement for some Riemann form. Now use that
an affine algebraic group with a faithful semisimple representation over a perfect field is
reductive (iAG, 21.60).22                                                                2

C OROLLARY 4.7 An abelian variety has complex multiplication if and only if its Mumford-
Tate group is commutative (in which case, it is a torus).
   22 Let W be a simple subrepresentation of V and U a normal unipotent subgroup of H: Because U is normal,

W U is stable under H , and because U is unipotent, W U ¤ 0. Therefore W U D W , and it follows that U D 1.
CHAPTER I. ANALYTIC THEORY                                                               37


P ROOF. The only connected commutative reductive groups are tori, and so this follows
from Proposition 4.2.                                                               2

P ROPOSITION 4.8 For abelian varieties A1 ; : : : ; An ,

                        MT.A1      An /  MT.A1 /      MT.An /;

and the projections MT.A1      An / ! MT.Ai / are surjective.

P ROOF. To get the inclusion, we identify the two vector spaces

                   H1 .A1      An ; Q/ ' H1 .A1 ; Q/ ˚    ˚ H1 .An ; Q/:

Then the statements are obvious from the definition of the Mumford-Tate group.            2


Infinity types
Let ˝ be a Galois extension of Q (for example, ˝ D Qal ). Recall that a complex con-
jugation on ˝ is any involution defined by complex conjugation on C and an embedding
˝ ,! C.
    Let K be a number field. The set of maps Hom.K; ˝/ ! Z is an abelian group, which
we denote by ZHom.K;˝/ . We sometimes regard ZHom.K;˝/ as the free abelian group on
Hom.K; ˝/, and regard f W Hom.K; ˝/ ! Z as a finite sum
                             X
                                        f ./; f ./ 2 Z:
                                      WK!˝

We let  2 Gal.˝=Q/ act on ZHom.K;Q/ according to the rule
                          X                    X
                    f D       f ./   ı  D    f . 1 ı /  :
                                                         

P ROPOSITION 4.9 Let K be a number field, and assume ˝ contains all conjugates of K.
Let D Hom.K; ˝/. The following three conditions on a map f W ! Z are equivalent:
  (a) f ./Cf .ı/ is constant (independent of W K ! ˝ and of the complex conjugation
       on ˝);
  (b) let K 0 be the composite of the CM-subfields23 of K, and fix a complex conjugation 
      on ˝; then
         i) f ./ depends only on jK 0 and
        ii) f ./ C f . ı / is constant (independent of ).
  (c) for a fixed complex conjugation  on ˝, and for all  2 Gal.˝=Q/;

                                .       1/. C 1/f D 0 D . C 1/.   1/f:             (31)

P ROOF. (a) H) (b). We may replace ˝ with the composite of the conjugates of K in ˝,
and so assume that ˝ is of finite degree. Suppose first that ˝ contains a CM-subfield, so
that K 0 is the largest such subfield. For any W K ! ˝, K 0 is the subfield of K fixed by
all commutators Œ; , where  2 Gal.˝=Q/ and  is the fixed complex conjugation on ˝
  23 Thus, K D Q if there are no CM-subfields of K.
CHAPTER I. ANALYTIC THEORY                                                                          38


(see 1.6). Hence two embeddings K ! ˝ agree on K 0 if and only if they differ by such a
commutator.24
    We are given that

          f ./ C f ./ D f ./ C f .  1 /;         all W K ! ˝;           2 Gal.˝=Q/:

On replacing  with  in this, we find that

                   f ./ D f .Œ;  ı /;     all W K ! ˝;          2 Gal.˝=Q/;

and so f ./ depends only on jK 0 .
    Next suppose that ˝ doesn’t contain a CM-field, and let K 0 be the largest totally real
subfield. The same argument as above shows that f ./ depends only on jK 0 , and so
2f ./ D f ./ C f . ı / Dconstant. Hence f ./ is independent of .
    (c) H) (b). The two equalities can be rewritten as

                                      .      C      1/f D 0                                   (32)
                                      .      C      1/f D 0:

Their difference gives
                                f D f for all  2 Gal.˝=K/:                                   (33)
When evaluated at , the (32) becomes

                             f . 1 / C f . 1 / D f ./ C f ./:                             (34)

Equation (33) shows that f ./ depends only on jK 0 (as in the proof of (a) H) (b)), and
(34) shows that f ./ C f ./ is independent of .
     (b) H) (a,c). Suppose first that K is a CM-field. Then (b)(ii) implies (a), because
 ı  D  ı E for every complex conjugation  of ˝. Similarly, it implies (c).
     Next suppose that K isn’t a CM-field. Then (b)(i) says f arises by extension from a
function on K 0 , for which (a) and (c) hold. It follows that (a) and (c) hold for f itself. 2

D EFINITION 4.10 An infinity type on K with values in ˝ is an element f 2 ZHom.K;˝/
satisfying the equivalent conditions of (4.9).

    The negative of the constant value f ./ C f . ı / is called the weight of f :

                              w.f / D f ./          f . ı / for all ; i .

Note that the weight is additive
                                           P            P
                                      w.     i fi / D     i w.fi /

and that a CM-type on a field K is exactly an infinity type of weight 1.
   We now write I.K/ for the group of infinity types with values in Qal .
   24 Let ; 0 W K ! ˝ be two embeddings. The obvious isomorphism K ! 0 K extends to an automorphism

 of ˝, and  and 0 agree on K 0 if and only if  fixes K 0 , i.e.,  D Œ;  for some  2 Gal.˝=Q/.
CHAPTER I. ANALYTIC THEORY                                                                      39


P ROPOSITION 4.11 Let K be a number field, and let K 0 be the composite of the CM-
subfields of K. Then        X
                       f 7!    f .jK 0 /W I.K 0 / ! I.K/
is an isomorphism. In particular, if K does not contain a CM-field, then

                                         Z D I.Q/ ! I.K/

is an isomorphism.

P ROOF. Immediate from the description of I.K/ given in (4.9b).                                  2

    It remains to determine I.K/ in the case that K is CM.

P ROPOSITION 4.12 Let K be a CM-field, and let f'1 ; : : : ; 'g g be a CM-type on K. Define
CM-types
                                 X
                       i D 'i C         ı 'j ; i D 1; : : : ; n;
                                   j ¤i
                            X
                       N D           ı 'j :
                                        1j n

Then f1 ; : : : ; n; N
                      g is a basis for the Z-module I.K/.

P ROOF. Let f 2 I.K/. Then
                        X                                      X                   
                  f D                 f .'i /i       w.f / C                         N
                                                                             f .'i / ;
                           1j n                                   1i n

                                                                                                   N
because the two sides agree on each 'i and have the same weight. This shows that f1 ; : : : ; n; g
spans I.K/, and it is obvious that it is linearly independent.                                 2

P ROPOSITION 4.13 For any CM-field K, there is a commutative diagram

                                                  1C
                                   I.K/                         I.K/

                                           w
                                                  Z

where the unmarked arrow sends m 2 Z to the constant function with value m.

P ROOF. For any f 2 I.K/ and W K ! Qal ;

                         ..1 C /f / ./ D f ./ C f ./ D w.f /;

i.e., .1 C /f is the constant function with value w.f /.                                        2

P ROPOSITION 4.14 For any CM-field K with totally real subfield F , there is an exact
sequence
                            f 7!                  al        .f;m/7!                 al
         0     ! I.K/              ! ZHom.K;Q /  Z                  ! ZHom.F ;Q /        !0   (35)
                          .f;w.f //                         f jF Cm

P ROOF. Obvious.                                                                                 2
CHAPTER I. ANALYTIC THEORY                                                                   40


   Consider number fields K  L. An infinity type f on K extends to an infinity type fL
on L by the rule:
                                 fL ./ D f .jL/:
Define
                                        I D lim I.K/
                                               !
(limit over all subfields of Q ; equivalently, over all CM-subfields of Qal ). In a natural way,
                              al

I can be identified with the group of locally constant homomorphisms f W Gal.Qal =Q/ ! Z
such that

             f ./ C f .  1 / is constant (independent of ;  2 Gal.Qal =Q/),

and with the group of locally constant homomorphism f W Gal.Qcm =Q/ ! Z such that

                f ./ C f ./ is constant (independent of  2 Gal.Qcm =Q/).

Such functions f are called infinity types on Qal or Qcm respectively.
R EMARK 4.15 Let K be a CM-field with largest totally real subfield F , and let f be an
infinity type on K. For a 2 F  and embedding W K ! Qal , .a/ D .a/, and so

                    .a/f ./  .a/f ./ D .a/f ./Cf ./ D .a/ w.f / :

Therefore,
                                      def
                                            Y
                                f .a/ D                   .a/f ./
                                               WK!Qal
                                            Y
                                      D                   .a/ w.f /
                                               WF !Qal

                                      D NmF =Q .a/ w.f / :
                                                                               
In particular, f maps OF     into f˙1g. The unit theorem (ANT, 5.1) shows that OF is of
                                                                          
finite index in OK , and so f is trivial on a subgroup of finite index in OK .

The Serre group
D EFINITION AND UNIVERSAL PROPERTY
For a number field K  Qal we define the Serre group S K of K to be the quotient of
.Gm /K=Q such that
                                                     al
                         X  .S K / D I.K/  ZHom.K;Q / :
More explicitly, S K is the quotient of .Gm /K=Q by its subgroup
                            \
                                       Ker.f W .Gm /K=Q ! Gm /:
                               f 2I.K/

It is an algebraic torus over Q.
R EMARK 4.16 As I.K/ is finitely generated, (4.15) shows that the kernel of

                                            K  ! S K .Q/                                  (36)
                        
contains a subgroup of OK  of finite index. We shall see later (4.22) that S K is the quotient
                                                                              
of .Gm /K=Q by the Zariski closure of any sufficiently small subgroup of OK     .
CHAPTER I. ANALYTIC THEORY                                                                 41


    Write 0 for the given inclusion of K into Qal . We let K denote the cocharacter of S K
that acts on characters as f 7! f .0 /.

L EMMA 4.17 Let T be an algebraic torus over Q. The following conditions on a cochar-
acter  of T are equivalent:
  (a) for every complex conjugation 0 of Qal , .0 C 1/ is defined over Q;
  (b)  is defined over some CM-subfield of Qal , and . C 1/ is defined over Q;
  (c) for all  2 Gal.Qal =Q/;

                              .      1/. C 1/ D 0 D . C 1/.     1/:                (37)

P ROOF. Similar to that of (4.9). (Note that requiring that . 1/. C 1/ D 0 for all  2
Gal.Qal =Q/ amounts to requiring that . C 1/ be defined over Q.)                      2

D EFINITION 4.18 Any one of the equivalent conditions in (4.17) will be called the Serre
condition.

    For an algebraic torus T that splits over a CM-field, the Serre condition simply says
that the weight
                                          def
                                   w./ D . C 1/
of  is defined over Q.

P ROPOSITION 4.19 The cocharacter K of S K satisfies the Serre condition. For any al-
gebraic torus T defined over Q and cocharacter  defined over K and satisfying the Serre
condition, there is a unique homomorphism W S K ! T (defined over Q) such that

                                         Qal ı K D :                                  (38)

P ROOF. Recall that

                           7! h; iW X .S K / ! Hom.X  .S K /; Z/

is an isomorphism of ZŒGal.Qal =Q/-modules. Because the characters of S K satisfy (31),
its cocharacters satisfy (37).
     Let T be a torus over Q, and let  2 X .T /. For  2 X  .T / and  2 Gal.Qal =Q/,
define
                                  f . / D h 1 ; i:
Then, for ;  2 Gal.Qal =Q/,
                                def
                      f . / D h 1 ; i D f . 1  / D .f /. /:

It follows that if T is split by K, then f . / depends only on jK. Moreover, if  satisfies
the Serre condition, then f is an infinity type on K. Therefore, we get a Gal.Qal =Q/-
equivariant homomorphism

                                   7! f W X  .T / ! X  .S K /;

which corresponds to a homomorphism

                                           W S K ! T:
CHAPTER I. ANALYTIC THEORY                                                                  42


For 0 the given inclusion of K into Qal ,
                                         f .0 / D h; i;
i.e.,
                                        hK ; f i D h; i;
which proves (38).                                                                           2


T HE NORM MAP.
P ROPOSITION 4.20 For any inclusion of number fields K  L, there is a unique homo-
morphism
                                NmL=K W S L ! S K
such that
                                        NmL=K ıL D K :
P ROOF. For an infinity type f on K, define fL . / D f .jK/. Then fL is an infinity type
on L, and the map f 7! fL W I.K/ ! I.L/ gives rise to the map NmL=K . Alternatively,
when L splits S K , we can apply Proposition 4.19.                                       2

        Therefore, an inclusion of number fields gives rise to a commutative diagram

                                     .Gm /L=Q           SL
                                          NmL=K           NmL=K                           (39)

                                     .Gm /K=Q           SK
        We define the Serre group .S; can / to be the inverse limit
                                     .S; can / D lim.S K ; K /;
where K runs over the subfields of Qal (or only the CM-subfields, see (4.23) below). The
character group of S is I , and  corresponds to f 7! f .1/.
P ROPOSITION 4.21 For any torus T defined over Q and cocharacter  satisfying the Serre
condition, there is a unique homomorphism W S ! T (defined over Q) such that
                                          Qal ı can D :
P ROOF. Apply Proposition 4.19.                                                              2


D ESCRIPTION OF THE S ERRE GROUP IN TERMS OF ARITHMETIC SUBGROUPS
Let K be a CM-field. We saw in (4.15) that the kernel of any infinity type f on K contains
                               
a subgroup of finite index in OK . As I.K/ is finitely generated, it follows that the kernel of
                                           K  ! S K .Q/
                                          
contains a subgroup N of finite index in OK . The next theorem shows the kernel of
              K
.Gm /K=Q ! S is the smallest algebraic subgroup of .Gm /K=Q containing any such an
N.
T HEOREM 4.22 The Serre group of K is the quotient of .Gm /K=Q by the Zariski closure
of any sufficiently small arithmetic subgroup of .Gm /K=Q .Q/ D K  .
P ROOF. To be added (cf. Serre 1968, pII-9, Exercise 1).                                     2
CHAPTER I. ANALYTIC THEORY                                                                      43


C ALCULATION OF THE S ERRE GROUP
P ROPOSITION 4.23 Let K  Qal be a number field, and let K 0 be the composite of the
CM-subfields of K. Then
                                                 0
                             NmK=K 0 W S K ! S K
is an isomorphism. In particular, if K doesn’t contain a CM-field, then

                                NmK=Q W S K ! S Q D Gm

is an isomorphism.

P ROOF. According to 4.9, the inclusion I.K 0 / ,! I.K/ is a bijection, which implies the
statement.                                                                              2


S OME EXACT SEQUENCES
For simplicity, from now on we take K  Qal to be a CM-field. There are homomorphisms

                                            X  ./
                         w K W Gm ! S K      f 7! w.f /W I.K/ ! Z
                         K W Gm ! SQKal     f 7! f .1/W I.K/ ! Z
                         t K W S K ! Gm      m 7! mW Z ! I.K/

Note that

                                          t ıw D 2
                                          t ı  D 1:

P ROPOSITION 4.24 For any CM-field K, the homomorphism NmK=Q W .Gm /K=Q ! Gm
factors through S K , and gives rise to a commutative diagram

                                               1C
                                SK                          SK
                                                                                               (40)
                                NmK=Q                      wK
                                               Gm

P ROOF. Apply Proposition 4.13.                                                                  2

    For a 2 K  , let Œa be the image of a in S K .Q/. For such points, the commutativity of
(40) becomes the commutativity of

                          1C                                                    1C
 K         S K .Q/                  S K .Q/           a        Œa                        Œa Na

              NmK=Q                  wK                           NmK=Q                   wK
                          Q                                                  NmK=Q .a/

P ROPOSITION 4.25 For any CM-field K and largest real subfield F , there is an exact se-
quence
                                 incl.
                                
                                NmK=Q                           .can.;w K /
               0 ! .Gm /F =Q          ! .Gm /E=Q  Gm                   ! SK ! 1               (41)
CHAPTER I. ANALYTIC THEORY                                                                 44


P ROOF. Apply Proposition 4.14.                                                             2

    In more detail, there is a commutative diagram

                                        1                          1

                                                     NmF=Q
        1         Ker               .Gm /F =Q                     Gm                1
                                                                    wK

        1         Ker               .Gm /K=Q                      SK                1


                                                       '
                              .Gm /K=Q =.Gm /F =Q            S K =w K .Gm /         1


                                        1                          1

from which (41) can be extracted.

Abelian varieties of CM-type
Let .V; J / be a rational Riemann pair. Then

                                    V ˝Q C D VC ˚ V ;

where V˙ are the ˙1 eigenspaces of J acting on V ˝Q C. Let  D .V;J / be the cocharacter
of GLVC such that .z/ acts as z on VC and as 1 on V . Then that .z/ acts as 1 on VC
and as z on V , and so  C  is the cocharacter sending z to the homothety v 7! zv. In
particular, it is defined over Q.
    When .V; J / is the rational Riemann pair of an abelian variety, we write A for .V;J / .

P ROPOSITION 4.26 Let K be a CM-subfield of Qal .
  (a) For an abelian variety A of CM-type, A is defined over K if and only if the re-
      flex field of A is contained in K, in which case there is a unique homomorphism
      A W S K ! MT.A/ such that A ı can D A .
  (b) For each abelian variety A of CM-type with reflex field contained in K, the homo-
      morphism A is surjective, and, as A varies, the A define an isomorphism K W S K !
      lim MT.A/.

P ROOF. (a) The maps A are surjective, and so S ! lim MT.A/ is surjective. It is injective
because the CM-types generate the character group of S .                                  2

C OROLLARY 4.27 For any abelian variety A of CM-type, there is a unique homomorphism
A W S ! MT.A/ such that A ı can D A . Each A is surjective, and the A realize S as
the inverse limit of the groups M T .A/.

P ROOF. Obvious from the proposition.                                                       2
CHAPTER I. ANALYTIC THEORY                                                                 45


5    Motives
The Hodge structure of an abelian variety
Discuss the Hodge structure on the cohomology of an abelian variety. Define the Hodge
classes. Show that the Mumford-Tate group is the subgroup fixing the Hodge classes.

Abelian motives
Use the Hodge classes to define a category of abelian motives (in a feeble sense), i.e., write
out the Grothendieck construction using the Hodge classes as correspondences.

Hodge structures
The usual stuff, including the Mumford-Tate group.

CM-motives
A rational Hodge structure is of CM-type if it is polarizable and its Mumford-Tate group is
commutative (hence a torus).

P ROPOSITION 5.1 A rational Hodge structure .V; h/ is of CM-type if and only if there
exists a homomorphism W S ! GLV such that C ı S D h (in which case  is unique).

P ROOF. O.K..                                                                               2

P ROPOSITION 5.2 The map .V; / 7! .V; R ı h/ defines an equivalence from the category
of representations of S to the category of Hodge structures of CM-type.

P ROOF. O.K..                                                                               2

P ROPOSITION 5.3 The functor HB defines an equivalence from the category of CM mo-
tives to the category of CM Hodge structures.

P ROOF. O.K..                                                                               2

A SIDE 5.4 There are similar results for the category of all abelian motives, but they are
much more complicated. Specifically, there is a pro-algebraic group G over Q together
with a cocharacter can such that
  ˘ the functor HB defines an equivalence from the category of abelian motives to the
      category of Hodge structures whose Mumford-Tate group is a quotient of .G; can /
      (abelian Hodge structures);
  ˘ the functor .V; / .V; R ı hcan / defines an equivalence from the category of repre-
      sentations of G to the category of abelian Hodge structures.
For a description of .G; can /, based on work of Deligne, Satake, et al, see Milne 1994.
Chapter II

The Arithmetic Theory

6     Abelian varieties and their good reductions
We now define an abelian variety over a field k to be a complete algebraic variety over k
together with a group structure defined by regular maps. For the following statements, see,
e.g., Milne 1986, 2.2, 7.1.
  ˘ Let A and B be abelian varieties. Every regular map A ! B sending 0A to 0B is
       a homomorphism. In particular, the group structure on A is uniquely determined by
       the zero element. It is commutative because a 7! a is a homomorphism.
    ˘ Every abelian variety is projective.

Complex abelian varieties and complex tori
The next proposition shows that, when k D C, the definition of “abelian variety” in this
chapter essentially agrees with that in the last chapter.

P ROPOSITION 6.1 For any abelian variety A over C, A.C/ is a complex torus admitting a
Riemann form. The functor A A.C/ defines an equivalence from the category of abelian
varieties over C to the category of complex tori admitting a Riemann form.

P ROOF. (S KETCH ; SEE M UMFORD 1970, I 3, FOR THE DETAILS .) Clearly A.C/ is a com-
plex Lie group, which is compact and connected because A is complete and connected.
Therefore it is a complex torus (2.1).
    For abelian varieties A, B, the map Hom.A; B/ ! Hom.A.C/; B.C// is obviously in-
jective, and it is surjective by Chow’s theorem1 . It remains to show that the essential image
of the functor consists of the complex tori admitting a Riemann form.
    Let M ' V = be a complex torus (and its canonical uniformization). We have to show
that there exists an ample2 invertible sheaf L on M if and only if there exists Riemann form.
Recall that the isomorphism classes of invertible sheaves on a manifold (or variety) X are
classified by H 1 .X; O X /, and that the first chern class of an invertible sheaf is the image of
     1 Recall that the smooth case of Chow’s theorem says that every projective complex manifold has a unique

structure of a nonsingular projective algebraic variety, and every holomorphic map of projective complex man-
ifolds is regular for these structures (Shafarevich 1994, VIII 3.1).
     2 Recall that this means that, for some n > 0, the sections of L˝n give an embedding of M as a closed

complex submanifold (hence, subvariety) of projective space.




                                                    46
CHAPTER II. THE ARITHMETIC THEORY                                                                      47


its cohomology class under the boundary map

                                         H 1 .X; OX
                                                  
                                                    / ! H 1 .X; Z/

defined by the exponential sequence

                                                       e 2 i./
                                                             
                                     0 ! Z ! OX           ! OX ! 0:

     Let    be an alternating Z-bilinear form    ! Z such that

                                     .J u; J v/ D      .u; v/;       u; v 2 V;

and let
                                      .ujv/ D    .u; J v/          i .u; v/
be the corresponding hermitian form on V (so is a Riemann form if and only if .j/ is
positive definite). One can show that there exists a map

                                         ˛W  ! C ;      j˛.z/j D 1;

such that
                        ˛.u C v/ D e i .ujv/  ˛.u/  ˛.v/;             all u; v 2 ,              (42)
and for any such ˛,
                                                                            1
                             u 7! eu ,     eu .z/ D ˛.u/  e .zju/C 2 .uju/ ;

is a one-cocycle on V with coefficients in OV whose image under the boundary map
                                                               ^                  _
                             1                                        2
                         H       .A; OV / ! H 2 .A; Z/ '                 H1 .A; Z/

is . Let L. ; ˛/ be the invertible sheaf defined by this cocycle. The following hold.
      Theorem of Appell-Humbert: Every invertible sheaf on A is isomorphic to
      L. ; ˛/ for a uniquely determined pair . ; ˛/.
      Theorem of Lefschetz: If . j / is positive definite, then the space of holomor-
      phic sections of L. ; ˛/˝n gives an embedding of M as a closed complex
      submanifold of projective space for each n  3; conversely, if L. ; ˛/˝n gives
      such an embedding for some n > 0, then . j / is positive definite.
Thus, M admits an ample invertible sheaf if and only if it admits a Riemann form.                       2

R EMARK 6.2 Let A be an abelian variety with complex multiplication        by E over an alge-
                                                 3
                                                                    L
braically closed field k of characteristic zero. Then Tgt0 .A/ ' '2˚ k' (as an E ˝Q k-
module), where ˚  Hom.E; k/ and k' is a one-dimensional k-vector space on which E
acts through '. It follows from (18) that Hom.E; k/ D ˚ t ˚ for every complex conjuga-
tion  on k. In particular, this applies with k D C: a complex abelian variety A with complex
multiplication by E defines a subset ˚  Hom.E; C/ such that Hom.E; C/ D ˚ t   1 ˚
for all automorphisms  of C.
    3 Recall that this means that E is a Q-subalgebra of End0 .A/ of degree 2 dim A (not necessarily a CM-

algebra).
CHAPTER II. THE ARITHMETIC THEORY                                                                             48


E XERCISE 6.3 Let E be a number field, and let ˚ be a subset of Hom.E; C/. Show that
the following conditions on .E; ˚ / are equivalent:
                   an abelian variety A over C with complex multiplication by E such that
  (a) there exists L
       Tgt0 .A/ ' '2˚ C' (as an E ˝Q C-module);
  (b) Hom.E; C/ D ˚ t   1 ˚ for all automorphisms  of CI
  (c) there exists a CM-subfield E0 of E and a CM-type ˚0 on E0 such that ˚ D f' j
      'jE0 2 ˚0 g;
  (d) j˚j D ŒEW Q=2 and there exists a CM-subfield E0 of E such that no two of the ' in
      ˚ are complex conjugates on E0 .
(Cf. Shimura and Taniyama 1961, 5.2, Theorem 1.)

Specialization of abelian varieties
Let k  K be algebraically closed fields, and let X be a smooth complete variety over K.
A variety X0 over k is called a specialization of X if there exists a commutative diagram
with cartesian squares

                                  X                     X                   X0
                                                                                                            (43)
                              Spec.K/               Spec.R/              Spec.k/

in which
   ˘ R is a normal finitely generated k-subalgebra of K and
  ˘ X is flat of finite type over R and X0 is a smooth complete variety over k (when R
       is local, this is equivalent to X being a smooth proper R-scheme (Hartshorne 1977,
       III 10.2, . . . ).
Every X has a specialization to k: since the polynomials defining X have only finitely many
coefficients, X has a model over a subfield L of K that is finitely generated over k; this
model extends to a smooth proper model over Spec R for some normal finitely generated
k-algebra R with field of fractions L; according to the Hilbert Nullstellensatz, there is a
k-algebra homomorphism R ! k, and we can take X0 to be the base change of X .
    In the diagram (43), let p 2 X .R/ and let p and p0 be its images in X.K/ and X0 .k/.
Then Tgtp .X / is a free R-module of rank dim X, and the maps in (43) induce isomorphisms
                              '                                                '
                 Tgtp .X /        Tgtp .X / ˝R K;           Tgtp .X / ˝R k ! Tgtp0 .X0 /                    (44)

[Add explanation that tangent spaces commute with base change, at least for smooth maps.]

The good reduction of abelian varieties
In this section, I review in as elementary fashion as possible, the theory of the good reduc-
tion of abelian varieties. Since most results have been incorporated into the theory of Néron
models, those familiar with that theory4 can skip the explanations.
    4 By which I mean those who have read the proofs in either Néron 1964 or Bosch et al. 1990. Note that,

except for (6.12), the present results are used in Néron’s theory, and so to deduce them from the main statements
of that theory would be circular.
CHAPTER II. THE ARITHMETIC THEORY                                                                     49


    Let R be a discrete valuation ring with field of fractions K and residue field k. We say
that an abelian variety over K has good reduction if it is the generic fibre of an abelian
                                def
scheme5 A over R; then A0 D A ˝R k is an abelian variety over k:We assume K has
characteristic zero and that k is perfect (although, this is not really necessary).

P ROPOSITION 6.4 Let A be an abelian scheme over R with generic fibre A. For any
smooth R-scheme X, every morphism XK ! A extends uniquely to a morphism X ! A,
i.e., A represents the functor on smooth R-schemes X A.XK /.

P ROOF. This follows from the next lemma and the following consequence of the valuative
criterion of properness (Hartshorne 1977, II 4.7):6
       Let Y ! S be a proper map, and let X ! S be a smooth map; every S -
       morphism X r Z ! Y with Z of pure codimension one in X extends uniquely
       to an S-morphism X ! Y .

L EMMA 6.5 Let S D Spec R, with R a discrete valuation ring, and let G be a smooth group
scheme over S . For any smooth scheme X over S and rational map f W X 99K G, the set of
points where f is not defined has pure codimension one in X:

P ROOF. Note that the set where f is defined will be open in each fibre of X=S. Define F to
be the rational map X  X 99K G, .x; y/ 7! f .x/f .y/ 1 . If f is defined at x 2 X (meaning
in an open neighbourhood of x/, then F is defined at .x; x/ 2 X  X, and F .x; x/ D e.
Conversely, if F is defined at .x; x/, then it will be defined on .x; U / for some open set U .
After possibly replacing U a smaller open set, f will be defined on U , and so the formula
f .x/ D F .x; u/f .u/, u 2 U shows that f is defined at x. Thus f is defined at x if and
only if F is defined at .x; x/:
                                                      '
    The rational map F defines a map OG;e ! k.X  X /. Since F sends .x; x/ to e (if
defined at .x; x//, we see that F is defined at .x; x/ if and only if Im.'/  OXX;.x;x/ :
    Now X  X is smooth over S, and hence is normal. Thus the divisor of an element of
k.X  X/ is defined, and we have that

                OX X;p D ff 2 k.X  X / j f does not have a pole at pg:                              2


C OROLLARY 6.6 Let A and B be abelian varieties over K, and let A and B be abelian
schemes over R with generic fibres A and B respectively. The restriction map

                                    HomR .A; B/ ! HomK .A; B/

is a bijection. In particular, B (if it exists) is unique up to a unique isomorphism.

P ROOF. The proposition shows that the restriction map MorR .A; B/ ! MorK .A; B/ is
bijective, and it is easy to see that homomorphisms correspond to homomorphisms. 2

P ROPOSITION 6.7 Let W A ! A_ be a polarization on A; then the extension A ! A_ of
 is a polarization on A (and hence reduces to a polarization on A0 ).

P ROOF. See Artin 1986, 4.4, and Chai and Faltings 1990.                                               2
   5 An abelian scheme over a scheme S is a smooth proper scheme over S together with a group structure.
   6 Add a proof of the deduction in this footnote.
CHAPTER II. THE ARITHMETIC THEORY                                                              50


     An elliptic curve has good reduction at a prime ideal p of R if and only if it can be
defined by a homogeneous polynomial F .X0 ; X1 ; X2 / with coefficients in R which, when
read modulo p, defines an elliptic curve over R=p. In particular, this means that, when it
has good reduction, the elliptic curve extends to a smooth closed subscheme of P2R . We
now prove a similar result for abelian varieties.
     Recall (Mumford 1999, II 8, p. 127) that the specialization map W Pn .K al / ! Pn .k al /
is defined as follows: represent P 2 Pn .K al / by .a0 W    W an /, where each ai 2 Ral and not
all ai lie in mal ; then .P / is represented by .aN 0 W : : : W aN n /.

L EMMA 6.8 For any closed subvariety X of PnK , the Zariski closure X of X in PnR has the
property that
                      X \ PnK D X (intersection inside PnR ),
and X is the unique flat subscheme of PnR with this property; moreover,

                                     .X.K al // D X .k al /:

P ROOF. Let a be the homomogeneous ideal in KŒX0 ; : : : ; Xn  of polynomials zero on X,
and let
                             a0 D a \ RŒX0 ; : : : ; Xn :
Then
                                  X D Proj RŒX0 ; : : : ; Xn =a0
with its natural inclusion into PnR and map X ! X has the required properties (Mumford
1999, II 8, Proposition 2).                                                         2

P ROPOSITION 6.9 Let A be an abelian variety over K with good reduction. For a suitable
choice of a closed immersion A ,! PnK , the closure A of A in PnR is an abelian scheme.

P ROOF. Let A be an abelian scheme over R whose general fibre is A. According to 6.7,
there is a divisor D on A whose Zariski closure DN on A is ample. Let A ,! PnR be the
                            N and let A ,! Pn be its generic fibre. Then the closure of
closed immersion defined by D,                K
       n
A in PR is A.                                                                        2

P ROPOSITION 6.10 Let A be an abelian variety over K with good reduction. If OK is
henselian (for example, if K is complete), then for every m prime to the characteristic of k,
the specialization map defines an isomorphism A.K/m ! A.k/m .

P ROOF. Let Z=mZ denote the constant group scheme over R (disjoint union of copies of
Spec R indexed by f0; : : : ; m 1g). Then
                                              .6:5/
                A.R/m ' Hom.Z=mZ; A/ ' Hom.Z=mZ; A/ ' A.K/m :

The map mW A ! A is étale (e.g., Milne 1986, 20.7). Using this and Hensel’s lemma, one
shows that the kernel of A.R/ ! A.k/ is uniquely divisible by m, and so A.R/m ! A.k/m
is a bijection.                                                                       2

    Extend the valuation on K to an algebraic closure KN of K, and let I  D  Gal. N K=K/
                                                               al
be the inertia and decomposition groups. Then D=I ' Gal.k =k/, and the ring of integers
of KN I is henselian (if OK is henselian, then KN I is a maximal unramified extension K un of
K).
CHAPTER II. THE ARITHMETIC THEORY                                                                51


C OROLLARY 6.11 With the above notation, A.KN I /m D A.K al /m for all m prime to the
characteristic of k.

P ROOF. The group A.KN I /m ' A.k al /m has m2 dim A elements.                                    2

   Hence, when A has good reduction, the representation ` of Gal.K al =K/ on T` A is
unramified (i.e., trivial on the inertia subgroup I ) for ` different from the characteristic of
k.

T HEOREM 6.12 (N ÉRON CRITERION ) If T` A is unramified for some ` different from the
characteristic of k, then A has good reduction.

P ROOF. For elliptic curves, this was known to Ogg and Shafarevich, and the criterion is
often named after them. As Serre observed, for abelian varieties it follows fairly directly
from the existence of Néron models (Serre and Tate 1968, Theorem 1). Recall that a smooth
group scheme A over R with generic fibre A is a Néron model for A if it represents the
functor on smooth R-schemes X 7! A.XK /. Such a model always exists (Néron 1964,
Artin 1986, or Bosch et al. 1990); it is obviously unique (up to a unique isomorphism).
When A has good reduction, and so extends to an abelian scheme A over R, Proposition
6.4 shows that A is a Néron model for A; conversely, when the Néron model is proper, it is
an abelian scheme and so A has good reduction.
    Let A be the Néron model for A, and let A0 be its special fibre. For m distinct from the
characteristic of k, the reduction map defines an isomorphism

                                      A.K al /Im ' A.k al /m

(see 6.10). Thus, if I acts trivially on T` A, then

                       A.k al /`n ' A.K al /`n  .Z=`n Z/2 dim A ;    all n:                   (45)

Every commutative algebraic group, for example, A0 , has a composition series whose quo-
tients are a finite group F , a unipotent group U , a torus T , and an abelian variety B. Now
F .k al /.`/ is finite, U.k al /.`/ D 0, and T .k al /.`/  .Q` =Z` /dim T . Thus (45) can hold only
if A0 is an abelian variety, in which case A is proper.                                            2

    An abelian variety over K is said to have potential good reduction if it acquires good
reduction over a finite extension of K. According to (6.11) and (6.12), a necessary and
sufficient condition for this is that the image of the inertia group under ` is finite.

C OROLLARY 6.13 Suppose that the k is finite and that for some ` ¤ p D char k, the image
of ` is commutative. Then A has potential good reduction.

P ROOF. We may assume that K is a complete local field. Because the image of ` is com-
mutative, local class field theory (CFT, I, 1.1) shows that the image I of the inertia group
                                                                                     
in Aut.T` A/ is a quotient of OK  . But 1 C pOK is a pro-p-group of finite index in OK  , and
so I has a pro-p-subgroup P of finite index. On the other hand, 1 C `  End.T` A/ is a pro-
`-subgroup of Aut.T` A/ of finite index. As ` ¤ p, the subgroups P and 1 C `  End.T` A/
intersect trivially, and so P maps injectively into the finite group Aut.T` A=`T` A/. It fol-
lows that I is finite.                                                                      2

C OROLLARY 6.14 An abelian variety has good reduction if it is isogenous to an abelian
variety with good reduction.
CHAPTER II. THE ARITHMETIC THEORY                                                                        52


P ROOF. Suppose A is isogenous to an abelian variety B with good reduction. Then there
exists an isogeny A ! B, and any such isogeny defines an injective map T` A ! T` B. As
T` B is unramified, so also is T` A.                                                 2



7    Abelian varieties with complex multiplication
Definition of a CM abelian variety
Let A be an abelian variety over a field k. If k can be embedded in C, then End0 .A/ acts
faithfully on H1 .A.C/; Q/, which has dimension 2 dim A, and so (see 1.2),
                                     ŒEnd0 .A/W Qred  2 dim A:                                       (46)
In general, for ` ¤ char k, End0 .A/ ˝Q Q` acts faithfully on V` A (e.g., Milne 1986, 12.5),
which again implies (46). When equality holds in (46), we say that A has complex multi-
plication over k (or be an abelian variety of CM-type over k, or be a CM abelian variety
over k).
R EMARK 7.1 Although we are interested here only in the case that k has characteristic
zero, this definition makes sense also in characteristic p. A theorem of Tate shows that every
abelian variety over a finite field k has complex multiplication over k (Tate 1966), and a
theorem of Grothendieck shows that every abelian variety with complex multiplication over
an algebraically closed field k of characteristic p is isogenous to an abelian variety defined
over a finite field (Oort 1973, Yu 2004).
E XERCISE 7.2 Let ` be a prime different from char k. Show that A has complex multipli-
cation over k if and only if the centralizer of End0 .A/ in EndQ` .V` A/ is commutative, in
which case it equals C.A/ ˝Q Q` , where C.A/ is the centre of End0 .A/.

Complex multiplication by a Q-algebra
Let A be an abelian variety over a field k, and let E be an étale Q-subalgebra of End0 .A/.
Recall that
                                      .1:3/                    .46/
                             ŒEW Q  ŒEnd0 .A/W Qred  2 dim A:
Equalities hold throughout if and only if A has complex multiplication and E is maximal,
in which case we say that A has complex multiplication by E over k.7 More generally,
we say that .A; i/ is an abelian variety with complex multiplication by E over k if i is an
injective homomorphism from a Q-algebra E of degree 2 dim A into End0 .A/.8
    Let A have complex multiplication by E over k, and let
                                       R D OA D E \ End.A/:
Then R is an order in E, i.e., it is simultaneously a subring and a lattice in E. It is a subring
of OE with integral closure OE . The ring R doesn’t change under base field extension, i.e.,
for any field k 0  k,
                                       R D E \ End.Ak 0 /:
    7 Shimura frequently complicates things by defining a number field to be a subfield of C. Thus, let A be

a simple abelian variety with complex multiplication, and let E D End0 .A/. Where we write “A has complex
                                                             
multiplication by E”, Shimura chooses an isomorphism iW E ! E 0  C of E with a subfield E 0 of C, and
writes “.A; i 1 / has complex multiplication by E 0 ”.
    8 In particular, this means that i.1/ D id .
                                              A
CHAPTER II. THE ARITHMETIC THEORY                                                                       53


To see this, note that, because E=R is torsion, it suffices to show that End.Ak 0 /= End.A/ is
torsion free. But if ˛ is an endomorphism of A that becomes divisible by m over k 0 , then it
is divisible by m over k (because, to say that ˛ is divisible by m in End.A/ means that it is
zero on Am , as multiplication by m defines an isomorphism A=Am ! A; cf. Milne 1986,
12.6).
     Let g D dim A, and let ` be a prime not equal to char k. Then T` A is a Z` -module of
rank 2g and V` A is a Q` -vector space of dimension 2g. The action of R on T` A extends to
               def                            def
actions of R` D R ˝Z Z` on T` A and of E` D Q` ˝Q E on V` A.

P ROPOSITION 7.3 (a) The E` -module V` A is free of rank 1.
    (b) We have
                              R` D E` \ End.T` A/:

P ROOF. (a) We have already noted that E` acts faithfully on V` A, and this implies that V` A
is free of rank 1 (see 1.2).
     (b) Let ˛ be an element of E` such that ˛.T` A/  T` A. For some m, `m ˛ 2 R` , and
if ˇ 2 R is chosen to be very close `-adically to `m ˛, then ˇT` A  `m T` A, which means
that ˇ vanishes on A`m . Hence ˇ D `m ˛0 for some ˛0 2 End.A/ \ E D R. Now ˛ and ˛0
are close in E` ; in particular, we may suppose ˛ ˛0 2 R` , and so ˛ 2 R` .                 2

C OROLLARY 7.4 The commutants of R in EndQ` .V` A/, EndZ` .T` A/, End0 .A/, and End.A/
are, respectively, E` , R` , F , and R.

P ROOF. Any endomorphism of V` A commuting with R commutes with E` , and therefore
lies in E` , because of (7.3a).
     Any endomorphism of T` A commuting with R extends to an endomorphism of V` A
preserving T` A and commuting with R, and so lies in E` \ End.T` A/ D R` .
     Let C be the commutant of E in End0 .A/. Then E is a subalgebra of C , so ŒEW Q 
ŒC W Q, and C ˝Q Q` is contained in the commutant E` of E in End.V` A/, so ŒEW Q 
ŒC W Q. Thus E D C .
     Finally, the commutant R in End.A/ contains R and is contained in C \ End.A/ D
E \ End.A/ D R.                                                                       2

C OROLLARY 7.5 Let .A; i / have complex multiplication by E, and let R D i 1 .End.A//.
Then any endomorphism of A commuting with i.a/ for all a 2 R is of the form i.b/ for
some b 2 R.

P ROOF. Apply the preceding corollary to i.E/  End0 .A/:                                                2

R EMARK 7.6 If ` does not divide .OE W R/, then R` is a product of discrete valuation rings,
and T` A is a free R` -module of rank 1,9 but in general this need not be true (Serre and Tate
                                def Q                      def Q
1968, p. 502). Similarly, Tm A D `jm T` A is a free Rm D `jm R` -module of rank 1 if m
is relatively prime to .OE W R/. [Make this into a lemma + a remark.]
    9 If ` doesn’t divide .O W R/, then the exact sequence
                            E

                                       0 ! R ! OE ! OE =R ! 0;

                 Q Z` , gives an isomorphism R` ' OE ˝ Z` , which is a product of discrete valuation rings,
when tensored with
say, OE ˝ Z`  Ri . Let Mi denote Ri regarded
                                            P                                                  ! Ri . Then
                                                    as an R` -module through the projection R` P
every projective R` -module is isomorphic to mi Mi for some mi  0. In particular, T` A  mi Mi ; as
T` A ˝Z` Q` is free of rank one over R` ˝Z` Q` , each mi D 1.
CHAPTER II. THE ARITHMETIC THEORY                                                          54


Specialization
Let .A; i/ be an abelian variety with complex multiplication by a CM-algebra E
                                                                             Qover a field
k of characteristic zero. If k contains all conjugates of E, then Tgt0 .A/ ' '2˚ k' (as
an E ˝Q k-module), where ˚ is a set of Q-algebra homomorphisms E ,! k and k' is a
one-dimensional k-vector space on which a 2 E acts as '.a/ — we say that .A; i / is of
CM-type .E; ˚/. For any complex conjugation  on k,

                                   ˚ t ˚ D Hom.E; k/

(see 3.11; recall that a complex conjugation on k is any involution induced by an inclusion
k ,! C and complex conjugation on C). If k  K, then a CM-type on E with values in k
can be regarded as a CM-type on E with values in K.
P ROPOSITION 7.7 Every specialization of a pair .A; i / of CM-type .E; ˚ / is of CM-type
.E; ˚ /.
P ROOF. Obvious from the relation (44) between the tangent spaces. (Cf. Shimura and
Taniyama 1961, 12.4, Proposition 26, p. 109).                                     2


Rigidity
L EMMA 7.8 For any abelian variety A over an algebraically closed field, the torsion points
on A are Zariski dense in A.
P ROOF. Let B be the Zariski closure of the set of torsion points in A. Then B is a complete
algebraic subgroup of A, and so its identity component B ı is an abelian subvariety of A.
For any prime not dividing the index of B ı in B and distinct from the characteristic of k,
B ı has `2 dim A points of order `, which shows that dim B ı D dim A. Hence B ı D A       2

P ROPOSITION 7.9 Let k be an algebraically closed subfield of C. The functor A            AC
from abelian varieties over k to abelian varieties over C is fully faithful, and its essential
image contains all abelian varieties with complex multiplication.
P ROOF. We shall make repeated use of the obvious fact that the inclusion A.k/ ,! A.C/
induces an isomorphism on the torsion points (because each group has n2 dim A points of
order dividing n).
     FAITHFUL : Let f; gW A ! A0 be homomorphisms of abelian varieties over k. If fC D
gC , then f and g agree on A.C/tors D A.k/tors , and so f D g by (7.8).
     F ULL : Let A and A0 be abelian varieties over k, and let f W AC ! A0C be a homomor-
phism. For any automorphism  of C fixing k, f and f agree on A.C/tors D A.k/tors , and
therefore on AC . This implies that f is defined over k (AG 16.9).
     E SSENTIAL IMAGE : Let A be a simple CM abelian variety over C, of CM-type .E; ˚ /
say. Then any specialization A0 of A to k is again of CM-type .E; ˚ / (see 7.7) and so A0C
is isogenous to A (see 3.13).
     Now consider an arbitrary CM abelian variety A over C. It follows from the last para-
graph that there exists an abelian variety A0 over k and an isogeny f W A0C ! A. The kernel
of f is a finite subgroup N of A0 .C/tors D A0 .k/tors . Let A00 D A0 =N . Then f defines an
isomorphism A00C ! A.                                                                      2

C OROLLARY 7.10 The functor A AC defines an equivalence from the category of CM
abelian varieties over k to the category of CM abelian varieties over C.
CHAPTER II. THE ARITHMETIC THEORY                                                         55


P ROOF. Clearly A has complex multiplication if and only if AC has complex multiplica-
tion, and so this follows from the proposition.                                     2

   In particular, an abelian variety with complex multiplication over C has a model over
any algebraically closed subfield of C which is unique up to a unique isomorphism.

P ROPOSITION 7.11 Let A be an abelian variety over k  C with complex multiplication
over k al . If A has complex multiplication over k, then k contains the reflex field of A;
conversely, if k contains the reflex field and A is simple, then A has complex multiplication
over k.

P ROOF. If A has complex multiplication over   Q k, then it has complex multiplication by a
CM-algebra E over k. Write Tgt0 .AC / ' '2˚ C' , where ˚ is a CM-type on E and
C' is a one-dimensional C-vector space on which E acts through '. Then Tgt0 .A/ is an
E ˝Q k-module such that Tgt0 .A/ ˝k C ' Tgt0 .AC /, and so the action of E on Tgt0 .A/
satisfies the condition (5) of Proposition 1.21, which implies that k  E  . L
    Now assume that A is simple. Let E D End0 .AkN /, and let Tgt0 .k/ N '         N
                                                                               '2˚ k' (usual
notation) with kN the algebraic closure of k in C and ˚  Hom.E; k/. N Because A is simple,
.E; ˚ / is primitive (3.13). The group Gal. N k=k/ acts on E, and we have to show that this
                                           N
action is trivial if k  E . Let  2 Gal.k=k/ act on E as  0 . One checks that  ˚ D ˚  0 .
                          

If k  E  , then ˚ D ˚ (see 1.17), and so ˚  0 D ˚. For any extension  00 of  0 to a
CM-field E1 containing E and Galois over Q, this implies that  00 fixes E by (1.10).     2


Good reduction
P ROPOSITION 7.12 Let A be an abelian variety over a number field k with complex mul-
tiplication. Then A has potential good reduction at all finite primes of k.

P ROOF. After possibly extending k, we may suppose that A has complex multiplication
                       def
by E over k. Let R D E \ End.A/ and let ` W Gal.Qal =k/ ! Aut.T` A/ be the `-adic
representation defined by A for some prime `. Because the elements of R are defined over
k, they commute with the action of Gal.Qal =k/. Therefore the image of ` is contained
in the centralizer of R in EndZ` .T` A/, which is R ˝Z Z` (see 7.4). In particular, it is
commutative, and this shows that A has potential good reduction at all primes of k not
dividing ` (apply 6.13).                                                               2


The degrees of isogenies
An isogeny ˛W A ! B defines a homomorphism ˛  W k.B/ ! k.A/, and the degree of ˛ is
defined to be Œk.A/W ˛  k.B/.
    The order of a finite group scheme N D Spec.R/ over a field k is the dimension of R
as a k-vector space.

L EMMA 7.13 The kernel of an isogeny of abelian varieties is a finite group scheme of order
equal to the degree of the isogeny.

P ROOF. Let ˛W A ! B be an isogeny. Then (e.g., Milne 1986, 8.1) ˛ OA is a locally free
OB -module, of rank r say. The fibre of ˛ OA at 0B is the affine ring of Ker.˛/, which
therefore is finite of order r. The fibre of ˛ OA at the generic point of B is k.A/, and so
r D Œk.A/W ˛  k.B/ D deg.˛/.                                                             2
CHAPTER II. THE ARITHMETIC THEORY                                                                   56


L EMMA 7.14 Let  be a free Z-module of finite rank and ˛W  !  a Z-linear map with
nonzero determinant.
  (a) The cokernel of ˛ is finite, with order equal to j det.˛/j.
  (b) The kernel of the map
                                          Q˛W . ˝ Q/= ! . ˝ Q/=
       defined by ˛ is finite with order equal to j det.˛/j.

P ROOF. (a) As ˛ has nonzero determinant, it maps  isomorphically onto a subgroup
˛./ of . According to a basic structure theorem, there exist bases e10 ; : : : ; em
                                                                                    0 for 
     00           00
and e1 ; : : : ; em for ˛./ such that

                          ei00 D ni ei0    ni 2 Z;    ni > 0;      i D 1; : : : ; n:

Let ei D ˛ 1 .ei00 /. Then e1 ; : : : ; en is a basis for , and

                        ˛.ei / D ni ei0     ni 2 Z;    ni > 0;       i D 1; : : : ; n:

Therefore the cokernel of ˛ is finite, with order equal to n1    nm . The matrix of ˛ with
respect to the bases e1 ; : : : em and e10 ; : : : em
                                                    0 is diag.n ; : : : ; n /. As the transition matrix
                                                               1           m
from one basis of  to a second has determinant ˙1, we see that the

                        det.˛/ D ˙ det.diag.n1 ; : : : ; nm // D ˙n1    nm :

    (b) Consider the commutative diagram,

                   0                       ˝Q             . ˝ Q/=                    0
                                 ˛              ˛˝1                   ˛Q

                   0                       ˝Q             . ˝ Q/=                    0:

Because det.˛/ ¤ 0, the map ˛ ˝ 1 is an isomorphism. Therefore the snake lemma gives
an isomorphism
                                Ker. Q˛/ ! Coker.˛/;
and so (b) follows from (a).                                                                         2

L EMMA 7.15 Let A be an abelian variety with complex multiplication by E, and let R D
E \ End.A/. An element of R is an isogeny if and only if it is not a zero-divisor,
                                                        0
     Q. Recall that E is an étale Q-subalgebra of End .A/, i.e., a product of fields, say
P ROOF
E D Ei . Obviously E D R ˝Z Q, and R  E. An element ˛ D .˛i / of R is not a zero-
divisor if and only if each component ˛i of ˛ is nonzero, or, equivalently, ˛ is an invertible
element of E.                                                                               2

P ROPOSITION 7.16 Let A be an abelian variety with complex multiplication by E, and let
R D End.A/ \ E. An element ˛ of R that is not a zero-divisor is an isogeny of degree
.RW ˛R/.
CHAPTER II. THE ARITHMETIC THEORY                                                                           57


P ROOF. If ˛ is not a zero-divisor, then it is invertible in E ' R ˝Z Q, and so it is an
isogeny. Let d be its degree, and choose a prime ` not dividing d  char.k/. Then d is the
determinant of ˛ acting on V` A (e.g., Milne 1986, 12.9).10 As V` A is free of rank 1 over
    def
E` D E ˝Q Q` , this determinant is equal to NmE` =Q` .˛/, which equals NmE=Q .˛/. But R
is a lattice in E, and so (a) of Lemma 7.14 shows that this norm equals .RW ˛R/.         2

P ROPOSITION 7.17 (S HIMURA AND TANIYAMA 1961, I, 2.8, T HM 1) Let k be an alge-
braically closed field of characteristic p > 0, and let ˛W A ! B be an isogeny of abelian
varieties over k. Assume that ˛  .k.B//  k.A/q for some power q D p m of p, and let d
be the dimension of the kernel of Tgt0 .˛/W Tgt0 .A/ ! Tgt0 .B/; then

                                               deg.˛/  q d .

    We offer two proofs, according to the taste and knowledge of the reader.

P ROOF OF (7.17) IN TERMS OF VARIETIES AND DIFFERENTIALS
L EMMA 7.18 Let L=K be a finitely generated extension of fields of characterstic p > 0
such that K  Lq for some power q of p. Then
                                                                1
                                           ŒLW K  q dim ˝L=K :
                                                                                                       q
P ROOF. Let x1 ; : : : ; xn be a minimal set of generators for L over K. Because xi 2 K,
                                              1
ŒLW K < q n , and it remains to prove dim ˝L=K    n. For each i , the extension

                                   L=K.x1 ; : : : ; xi 1 ; xi C1 ; : : : ; xn /

is nontrivial and purely inseparable because L  K  Lq . There therefore exists a K-
derivation of Di of L such that Di .xi / ¤ 0 but Di .xj / D 0 for j ¤ i , namely, @x@ . The
                                                                                     i
Di are linearly independent, from which the conclusion follows.                           2

P ROOF ( OF 7.17) In the lemma, take L D k.A/ and K D ˛  .k.B//. Then

                                        deg.˛/ D ŒLW K
                                         1
                                    dim ˝L=K D dim Ker.Tgt0 .˛//;

and so the proposition follows from the lemma.                                                                2


P ROOF OF (7.17) IN TERMS OF FINITE GROUP SCHEMES
The condition on ˛ implies that Ker.˛/ is connected, and therefore its affine ring is of the
                           p r1          p rs
form kŒT1 ; : : : ; Ts =.T1 ; : : : ; Ts / for some family .ri /1i s of integers ri  1 (iAG,
11.29). Let q D p m . Then each ri  m because ˛  .k.B//  k.A/q , and

                        s D dimk Tgt0 .Ker.˛// D dimk Ker.Tgt0 .˛// D d:

Therefore,                                     Ys
                                  deg.˛/ D                p ri  p ms D q d :
                                                   i D1
   10 When k has characteristic zero, we may suppose that k D C, and write A D Cg =.           The order of the
kernel of ˛Q is the degree of ˛ (7.13), which equals the determinant of ˛ acting on  ˝ Q by (b) of Lemma 7.14.
But V` A D  ˝ Q` , and so this is also the determinant of ˛ acting on V` A.
CHAPTER II. THE ARITHMETIC THEORY                                                           58


a-multiplications: first approach
Let A be an abelian variety with complex multiplication by E over a field k, and let R D
E \ End.A/. An element of R is an isogeny if and only if it is not a zero-divisor, and an
ideal a in R contains an isogeny if and only if it is a lattice in E — we call ideals with this
property lattice ideals. We wish to attach to each lattice ideal a in R an isogeny a W A ! Aa
with certain properties. The shortest definition is to take Aa to be the quotient of A by the
finite group scheme                            \
                                    Ker.a/ D       Ker.a/:
                                                       a2a

However, the formation of quotients by finite group schemes in characteristic p is subtle
(Mumford 1970, p. 109–123)11 , and was certainly not available to Shimura and Taniyama.
In this subsection, we give the original elementary definition, and in the next, we give a
functorial definition.

D EFINITION 7.19 Let A be an abelian variety with complex multiplication by E over a
field k, and let a be a lattice ideal in R. A surjective homomorphism a W A ! Aa is an
a-multiplication if every homomorphism aW A ! A with a 2 a factors through a , and a is
universal for this property, in the sense that, for every surjective homomorphism 0 W A ! A0
with the same property, there is a homomorphism ˛W A0 ! Aa , necessarily unique, such that
˛ ı 0 D a :
                                                      Aa
                                              a
                                                             9Š
                                                   a
                                      A                       A     9Š˛

                                                             9Š
                                              0
                                                             A0 :
    An abelian variety B for which there exists an a-multiplication A ! B is called an a-
transform of A.

E XAMPLE 7.20 (a) If a is principal, say, a D .a/, then aW A ! A is an a-multiplication
(obvious from the definition) — this explains the name “a-multiplication”. More generally,
if W A ! A0 is an a-multiplication, then
                                               a         
                                           A ! A ! A0

is an aa-multiplication for any a 2 E such that aa  R (obvious from the construction in
7.22 below).
     (b) Let .E; ˚/ be a CM-pair, and let A D C˚ =˚./ for some lattice  in E. For any
                     def
lattice ideal a in R D End.A/ \ E,

                          Ker.a/ D fz C ˚./ j az 2 ˚./ all a 2 ag
                                   ' a 1 =R;

where a 1 D fa 2 E j aa  Rg. The quotient map C˚ =˚./ ! C˚ =˚.a 1 / is an a-
multiplication.
  11 Compare the proof of (7.22) with that of Mumford 1970, III, Theorem 1, p. 111.
CHAPTER II. THE ARITHMETIC THEORY                                                                         59


R EMARK 7.21 (a) The universal property shows that an a-multiplication, if it exists, is
unique up to a unique isomorphism.
    (b) Let a 2 a be an isogeny; because a factors through a , the map a is an isogeny.
    (c) The universal property, applied to a ı a for a 2 R, shows that, Aa has complex
multiplication by E over k, and a is an E-isogeny. Moreover, R  End.Aa / \ E, but the
inclusion may be strict unless R D OE .12
    (d) If W A ! B is an a-multiplication, then so also is k 0 W Ak 0 ! Bk 0 for any k 0  k.
This follows from the construction in (7.22) below.

P ROPOSITION 7.22 An a-multiplication exists for each lattice ideal a:

P ROOF. Choose a set of generators a1 ; :::; an of a, and define Aa to be the image of

                                         x 7! .a1 x; : : :/W A ! An :                                   (47)
               P
For any a D       i ri ai 2 a, the diagram
                                           0a 1
                                              1
                                           @ :: A
                                              :
                                            an            .r1 ; ;rn /
                                     A               An                    A
                                                      a

shows that aW A ! A factors through a .
   Let 0 W A ! A0 be a quotient map such that each ai factors through 0 , say, ˛i ı 0 D ai .
Then the composite of
                                                             0˛ 1
                                                                 1
                                                       ˛D@ :: A
                                                            :
                                             0           ˛n
                                    A               A0                     An                           (48)
is x 7! .a1 x; : : :/W A ! An , which shows that ˛ ı 0 D a .                                              2

R EMARK 7.23 A surjective homomorphism W A ! B is an a-multiplication if and only if
every homomorphism aW A ! A defined by an element of a factors through  and one (hence
every) family .ai /1i n of generators for a defines an isomorphism of B onto the image
of A in An . Alternatively, a surjective homomorphism W A ! B is an a-multiplication if it
maps k.B/ isomorphically onto the composite of the fields a k.A/ for a 2 a — this is the
original definition (Shimura and Taniyama 1961, 7.1).

P ROPOSITION 7.24 Let A be an abelian variety with complex multiplication by E over k,
and let W A ! B and 0 W A ! B 0 be a and a0 -multiplications. There exists an E-isogeny
˛W B ! B 0 such that ˛ ı  D 0 if only if a  a0 :

P ROOF. If a  a0 , then aW A ! A factors through  when a 2 a0 , and so ˛ exists by the
                                                                                         0
universality of 0 . For the converse, note that there are natural (projection) maps AaCa !
       0                                                       0            0
Aa ; Aa . If there exists an E-isogeny ˛ such that ˛ ı a D a , then AaCa ! Aa is injective,
which implies that a C a0 D a by (7.29) below.13                                           2
   12 Over C (at least), A is E-isogenous to an abelian variety with End.A/ \ E D O (see 3.9), but every such
                                                                                   E
isogeny is an a-multiplication for some a (see below).
   13 Small problem here: need to check a C a0 is projective (or else assume R D O , which is all we really
                                                                                    E
need). However, it seems to me that one should be able to prove (7.24) without counting degrees. Note that, in
order to prove (7.24), it suffices to prove (7.26).
CHAPTER II. THE ARITHMETIC THEORY                                                                       60


C OROLLARY 7.25 Let W A ! B and 0 W A ! B 0 be a and a0 -multiplications; if there exists
an E-isomorphism ˛W B ! B 0 such that ˛ ı  D 0 , then a D a0 .

P ROOF. The existence of ˛ implies that a  a0 , and the existence of its inverse implies that
a0  a.                                                                                     2

C OROLLARY 7.26 Let a 2 End.A/ \ E. If aW A ! A factors through an a-multiplication,
then a 2 a.

P ROOF. The map aW A ! A is an .a/-multiplication, and so if there exists an E-isogeny ˛
such that ˛ ı a D a, then a  .a/.                                                    2

R EMARK 7.27 Let W A ! B be an a-multiplication. Let a1 ; : : : ; an be a basis for a, and
let ai D ˛i ı . In the diagram

                                                                 ˛1                    a1
                                                                        !                    !
                                   ˛                              ::                   ::
                A            B             An          ˛D           :            aD      :       ;
                                                                 ˛n                    an
                             a

˛ maps B isomorphically onto the image of a. For any prime ` different from the charac-
teristic of k, we get a diagram

                                            T`             T` ˛
                                    T` A           T` B                 T` A n
                                                    T` a

in which T` ˛ maps T` B isomorphically onto the image of T` a.

P ROPOSITION 7.28 If W A ! A0 is an a-multiplication, and 0 W A0 ! A00 is an a0 -multiplication,
then 0 ı  is an a0 a-multiplication.

P ROOF. Let a D .a1 ; :::; am /, and let a0 D .a10 ; :::; am
                                                           0 /; then a0 a D .: : : ; a0 a ; : : :/, and one
                                                                                      i j
can show that A is isomorphic to the image of A under x 7! .: : : ; ai0 aj x; : : :/ (alternatively,
                00

use (7.38) and (49)).                                                                                     2

P ROPOSITION 7.29 For any a-multiplication , deg./ D .OE W a/ provided a is invertible
(locally free of rank 1).

P ROOF. When a is principal, a D .a/, we proved this in (7.16). Next note that if a prime
l doesn’t divide .RW a/, then it doesn’t divide deg a . For, by the Chinese remainder the-
                                                                    14
                  an a 2.a/a such that l doesn’t divide .RW .a//; write .a/ D ab; then
orem, there exists
      a        b
deg . / deg T D deg  D .OE W .a//.
    Let a D q be the primary decomposition of a. Because the       Q nonzero primeQideals
in R are maximal, the q are relatively prime. Therefore, a D q, and R=a ' R=q
(Chinese remainder theorem). As a is projective, R=a has projective dimension  1; it
follows that each R=q has projective dimension  1, and so q is projective. After (7.28),
   14 If l doesn’t divide .RW a/, then R D a C .l/, and so there exists an a 2 O such that
                                                                                E

                                                a0     mod a
                                                a1     mod l:
CHAPTER II. THE ARITHMETIC THEORY                                                                          61


we may therefore suppose a to be primary for some prime ideal p. Let p \ Z D .p/. The
localization Rp of R relative to the multiplicative set R r .p/ is semilocal and noetherian.
                def
The ideal ap D aRp is projective, and therefore free, say ap D .a=s/ with a 2 a and s 2 Z
not divisible by p. Then .a/ D ab with b an ideal such that b \ Z is prime to p. As a is
p-primary, some power of p lies in a and so deg.a / and .RW a/ are both powers of p. On
the other hand, deg.b / and .RW b/ are both relatively prime to p. As

                           deg.a /  deg.b / D deg..a/ /
                                 .RW a/.RW b/ D .RW .a//         (as a C b D R)

and
                                                     .7:16/
                                        deg..a/ / D .RW .a//
the statement follows.15                                                                                     2

C OROLLARY 7.30 Let a be an invertible ideal in R. An E-isogeny W A ! B is an a-
multiplication if and only if deg./ D .RW a/ and the maps aW A ! A for a 2 a factor through
.

P ROOF. We only have to prove the sufficiency of the conditions. According to the definition
(7.19), there exists an E-isogeny ˛W B ! Aa such that ˛ ı  D a . Then deg.˛/ deg./ D
deg.a /, and so ˛ is an isogeny of degree 1, i.e., an isomorphism.                        2

P ROPOSITION 7.31 Let E be a CM-algebra, and let A and B be abelian varieties with
complex multiplication by E over an algebraically closed field k of characteristic zero. If
A and B are E-isogenous, then there exists a lattice ideal a and an a-multiplication A ! B.

P ROOF. After (7.10), it suffices to prove this for k D C. Because A and B are E-isogenous,
they have the same type ˚. After choosing E-basis elements for H1 .A; Q/ and H1 .B; Q/,
we have isomorphisms

                             C˚ =˚.a/ ! A.C/;            C˚ =˚.b/ ! B.C/:

Changing the choice of basis elements changes the ideals by principal ideals, and so we may
suppose that a  b. The quotient map C˚ =˚.a/ ! C˚ =˚.b/ is an ab 1 -multiplication. 2

P ROPOSITION 7.32 Let A be an abelian variety with multiplication by E over a number
                                                                                     def
field k, and assume that A has good reduction at a prime p of k. The reduction to k0 D Ok =p
of any a-multiplication W A ! B is again an a-multiplication.

P ROOF. Let a1 ; : : : ; an be a basis for a, and let ai D ˛i ı . In the diagram

                                                                ˛1                     a1
                                                                      !                      !
                                   ˛                            ::                     ::
                 A           B           An            ˛D         :            aD        :       ;
                                                                ˛n                     an
                             a
   15 When R D O
                    E , the proof is a little easier. According to the Chinese remainder theorem, there exists
an a 2 OE such that .a/ D ab with OE W a/ and .OE W b/ relatively prime. [Take a to be any element of OE
satisfying an appropriate congruence condition for each prime ideal p of OE such that .OE W p/ is not prime to
.OE W a/.] Then
                        deg.a / deg.b / D deg..a/ / D .OE W .a// D .OE W a/.OE W b/:
The only primes dividing deg.a / (resp. deg.b /) are those dividing .OE W a/ (resp. .OE W b/), and so we must
have deg.a / D .OE W a/ and deg.b / D .OW b/.
CHAPTER II. THE ARITHMETIC THEORY                                                         62


˛ maps B isomorphically onto the image of a. Let A and B be abelian schemes over Op
with general fibre A and B. Then the diagram extends uniquely to a diagram over Op
(see 6.6), and reduces to a similar diagram over k0 , which proves the proposition. (For an
alternative proof, see 7.34.)                                                             2


a-multiplications: second approach
In this subsection, R is a commutative ring.
P ROPOSITION 7.33 Let A be a commutative algebraic group A over a field k with an action
of R. For any finitely presented R-module M , the functor

                     AM .T / D HomR .M; A.T //          (T a k-scheme)

is represented by a commutative algebraic group AM over k with an action of R. Moreover,

                                   AM ˝R N ' .AM /N :                                   (49)

If M is projective and A is an abelian variety, then AM is an abelian variety (of dimension
r dim A if M is locally free of rank r).
P ROOF. If M D Rn , then AM is represented by An . The functor M            AM transforms
cokernels to kernels, and so a presentation

                                  Rm ! Rn ! M ! 0;

realizes AM as a kernel
                                  0 ! AM ! An ! Am :
Define AM to be the kernel in the sense of algebraic groups.
   For the second statement, use that there is an isomorphism of functors

                 HomR .N; HomR .M; A.T /// ' HomR .M ˝R N; A.T //:

     For the final statement, if M is projective, it is a direct summand of a free R-module
of finite rank. Thus AM is a direct factor of a product of copies of A, and so is an abelian
variety. Assume that M is of constant rank r. For an algebraic closure kN of k and a prime
` ¤ char k;
               N ` D HomR .M; A.k/
           AM .k/               N `/
                                                  def               def
                                    N ` /;
                    ' HomR` .M` ; A.k/         R` D Z` ˝ R;     M` D Z` ˝Z M .
                                                                                    N ` is
But M` is free of rank r over R` (because R is semi-local), and so the order of AM .k/
l 2r dim A         M
           . Thus A has dimension r dim A.                                              2

R EMARK 7.34 The proposition (and its proof) applies over an arbitrary base scheme S .
Moreover, the functor A AM commutes with base change (because A AM obviously
does). For example, if A is an abelian scheme over the ring of integers Ok in a local field k
and M is projective, then AM is an abelian scheme over Ok with general fibre .Ak /M .
P ROPOSITION 7.35 Let R act on an abelian variety A over a field k. For any finitely
presented R-module M and ` ¤ char k,
                                                 def               def
            T` .AM / ' HomR` .M` ; T` A/;      R` D Z` ˝ R;    M` D Z` ˝Z M .
CHAPTER II. THE ARITHMETIC THEORY                                                       63


P ROOF. As in the proof of (7.33),

                                k/`n ' HomR` .M` ; A. N
                            AM . N                   k/`n /:

Now pass to the inverse limit over n.                                                    2

   Let R D EndR .A/. For any R-linear map ˛W M ! R and a 2 A.T /, we get an element

                                  x 7! ˛.x/  aW M ! A.T /

of AM .T /. In this way, we get a map HomR .M; R/ ! HomR .A; AM /.

P ROPOSITION 7.36 If M is projective, then HomR .M; R/ ' HomR .A; AM /:

P ROOF. When M D R, the map is simply R ' EndR .A/. Similarly, when M D Rn , the
map is an isomorphism. In the general case, M ˚ N  Rn for some projective module N ,
and we have a commutative diagram

         HomR .M; R/ ˚ HomR .N; R/             HomR .A; AM / ˚ HomR .A; AN /
                        '                                       '
                                           '
                HomR .Rn ; R/                           HomR .A; An /:
                                                                                         2



P ROPOSITION 7.37 Let A be an abelian variety over a field k, and let R be a commutative
subring of End.A/ such that R ˝Z Q is a product of fields and ŒRW Z D 2 dim A. For any
invertible ideal a in R, the map a W A ! Aa corresponding to the inclusion a ,! A is an
                        def T
isogeny with kernel Aa D a2a Ker.a/.

P ROOF. The functor M       AM sends cokernels to kernels, and so the exact sequence

                                  0 ! a ! R ! R=a ! 0

gives rise to an exact sequence
                                                  a
                                  0 ! AR=a ! A ! Aa :

Clearly AR=a D Aa , and so it remains to show that a is surjective, but for a prime ` such
that a` D R` , T` .a / is an isomorphism by an earlier result, from which this follows.  2

C OROLLARY 7.38 Under the hypotheses of the proposition, the homomorphism

                                        a W A ! Aa

corresponding to the inclusion a ,! R is an a-multiplication.

P ROOF. A family of generators .ai /1i n for a defines an exact sequence

                                     Rm ! Rn ! a ! 0

and hence an exact sequence
                                   0 ! Aa ! An ! Am :
CHAPTER II. THE ARITHMETIC THEORY                                                           64


The composite of
                                               Rn ! a ! R
              P
is .ri / 7!       ri ai , and so the composite of

                                                a
                                              A ! Aa ,! An

is x 7! .ai x/1i n . As a is surjective, it follows that Aa maps onto the image of A in An ,
and so a is an a-multiplication (as shown in the proof of 7.22).                           2

R EMARK 7.39 Corollary 7.38 fails if a is not invertible. Then Aa need not be connected,
A ! .Aa /ı is the a-multiplication, and Aa =.Aa /ı ' Ext1R .R=a; A/ (see Waterhouse 1969,
Appendix).

a-multiplications: complements
                                                                     def
Let W A ! B be an a-multiplication, and let a 2 a 1 D fa 2 E j aa 2 Rg. Then  ı a 2
Hom.A; B/ (rather than Hom0 .A; B/). To see this, choose a basis for a1 ; : : : ; an for a, and
note that the composite of the “homomorphisms”

                                          a      x7!.:::;ai x;:::/
                                      A !A                     ! An

is a homomorphism into Aa  An .

P ROPOSITION 7.40 Let A have complex multiplication by E over k.
    (a) Let W A ! B be an a-multiplication. Then the map

                                   a 7! a ı aW a 1 ! HomR .A; B/

is an isomorphism. In particular, every R-isogeny A ! B is a b-multiplication for some
ideal b.
    (b) Assume that OE D End.A/ \ E. For any lattice ideals a  b in OE ,

                                       HomOE .Aa ; Ab / ' a 1 b.

    [Better to state this in terms of invertible modules.]

P ROOF. (a) In view of (7.38), the first statement is a special case of (7.36). For the second,
recall (7.20) that a ı a is an aa-multiplication.
                                  1
    (b) Recall that Ab ' .Aa /a b (see 7.28), and so this follows from (a).                  2

    In more down-to-earth terms, any two E-isogenies A ! B differ by an E-“isogeny”
A ! A, which is an element of E. When  is an a-multiplication, the elements of E such
that  ı a is an isogeny (no quotes) are exactly those in a 1 .

P ROPOSITION 7.41 Let A have complex multiplication by OE over an algebraically closed
field k of characteristic zero. Then a 7! Aa defines an isomorphism from the ideal class
group of OE to the set of isogeny classes of abelian varieties with complex multiplication
by OE over k with the same CM-type as A.
CHAPTER II. THE ARITHMETIC THEORY                                                                 65


P ROOF. Proposition 7.40 shows that every abelian variety isogenous to A is an a-transform
for some ideal a, and so the map is surjective. As aW A ! A is an .a/-multiplication, prin-
cipal ideals ideals map to A. Finally, if Aa is OE -isomorphic to A, then

                                 OE ' HomOE .A; Aa / ' a 1 ;

and so a is principal. [Better to state this in terms of invertible modules.]                      2

P ROPOSITION 7.42 Let A and B be abelian varieties with multiplication by OE over a
number field k, and assume that they have good reduction at a prime p of k. If A and B are
isogenous, every OE -isogeny W A0 ! B0 lifts to an a-multiplication W A ! B for some
lattice ideal a, possibly after a finite extension of k. In particular,  is an a-multiplication
(over a finite extension).

P ROOF. Since A and B are isogenous, there is an a-multiplication W A ! B for some
lattice ideal a by (7.31) (after a finite extension of k). According to Proposition 7.32,
0 W A0 ! B0 is also an a-multiplication. Hence the reduction map

                              HomOE .A; B/ ! HomOE .A0 ; B0 /

is an isomorphism because both are isomorphic to a 1 , via  and 0 respectively (7.40).
Therefore,  lifts to an isogeny 0 W A ! B, which is a b-multiplication (see 7.40).  2

N OTES Most of the definitions and results on a-multiplications go back to Shimura and Taniyama
1961, but sometimes with inessential hypotheses. The functorial definition 7.33 was used for elliptic
curves in Serre 1967, p. 294. Giraud (with Raynaud) extended 7.29 to the case of arbitrary orders R
in OE (Giraud 1968, Proposition 2, Remark 1).
CHAPTER II. THE ARITHMETIC THEORY                                                                        66


8     The Shimura-Taniyama formula
Review of numerical norms
                                                                            Q mi
Let K be a number field. The norm of a fractional ideal a D                  pi of K is defined by
                                        Y
                       NmK=Q .a/ D              .pi /mi f .pi =pi / ;   .pi / D pi \ Z:
                                            i

For a principal ideal a D .a/, NmK=Q .a/ is the principal ideal generated by NmK=Q a. The
numerical norm Na D NK=Q a of a nonzero ideal a in OK is defined to be .OK W a/. We
have
                                  .Na/ D NmK=Q .a/:
As NmK=Q is a homomorphism, so also is N,

                                    .OK W a/.OK W b/ D .OK W ab/:

For example, if p is prime and .p/ D p \ Z, then

                             NmK=Q p D .p f .p=p/ /;             Np D p f .p=p/ :

See ANT 4.1, 4.2. Similar statements apply to ideals in products of number fields. In
the next subsection, we sometimes do not distinguish a positive integer from the ideal it
generates.

Statement and proof
T HEOREM 8.1 Let A be an abelian variety with complex multiplication by a CM-algebra
E over a finite extension k of Qp . Suppose that k contains all conjugates of E and let P
be a prime ideal of Ok at which A has good reduction.16 Assume (i) that p is unramified in
E and (ii) that End.A/ \ E D OE .
  (a) There exists an element  2 OE inducing the Frobenius endomorphism on the re-
      duction of A.
    (b) The ideal generated by  factors as follows,
                                         Y
                                  ./ D          ' 1 .Nmk='E P/                                        (50)
                                                     '2˚

       where ˚  Hom.E; k/ is the CM-type of A.
                                                        def
P ROOF. Let A0 be the reduction of A to k0 D Ok =P, and let

                                  q D jk0 j D .Ok W P/ D p f .P=p/ :

    (a) Recall that the reduction map End.A/ ! End.A0 / is injective. The ring End.A0 / \
E is an order in E containing End.A/ \ E. As the latter is the maximal order OE , so must
be the former. The Frobenius endomorphism of A0 commutes with all endomorphisms of
A0 , and so the statement follows from (7.4).
    16 To recall, the hypotheses mean that E is a Q-subalgebra of End0 .A/ of degree 2 dim A, and that
                                                                    L
HomQ-alg .E; k/ has ŒEW Q elements. They imply that Tgt0 .A/ ' '2˚ k' as E ˝Q k-modules, where ˚
is a subset of HomQ-alg .E; k/ and k' is a one-dimensional k-vector space on which E acts through the map '.
CHAPTER II. THE ARITHMETIC THEORY                                                                         67


   (b) Let T D Tgt0 .A/. It is an E ˝L
                                     Q k-space, and T has a k-basis .e' /'2˚ such that
ae' D '.a/e' for all a 2 E, i.e., T ' '2˚ k' , where k' D k with E acting through '.
                                                                  def                  def
By assumption, A has a smooth model A over OP D Ok;P , and T D Tgt0 .A/ is such that
                                                                                 def
                           T ˝OP k ' T and T =PT ' Tgt0 .A0 / D T0

(see (44)).
     Because .p/ is unramified in OE , the ring OE ˝Z Zp is étale
                                                                 Q over Zp , and so OE ˝Z
OP is étale over OP . It follows  that the isomorphism E ˝Q k '  WE !k k induces an iso-
                                                                                    17
                            Q
morphism OE ˝Z OP '  WE        L!k O , where O D OP with OE acting throughL . Sim-
ilarly, the isomorphism T ' '2˚ k' induces an isomorphism T ˝Ok OP ' '2˚ O' ,
where O' D OP with OE acting through '. This means that the k-basis .e' /'2˚ for T
can be chosen to be an OP -basis for T ˝ OP . (Cf. Shimura 1998, 13.2.)
     Because  N D q, the ideal ./ is divisible only by primes dividing p, say,
                                          Y
                                  ./ D         pm v
                                                 v ;   mv  0;
                                                  vjp

For h the class number of E, let

                                        pvmv h D . v /;         v 2 OE ;                                (51)

and let

                                    ˚v D f' 2 ˚ j ' 1 .P/ D pv g;
                                    dv D j˚v j :

     The kernel of v W T0 ! T0 is the span of the e' for which '. v / 2 P, i.e., such that v
lies in the prime ideal ' 1 .P/. But pv is the only prime ideal in OE containing v , and so
'. v / 2 P if and only if ' 1 .P/ D pv W
                                              v
                                  Ker.T0 ! T0 / D he' j ' 2 ˚v i:
                                                                                                            h
Since  h W A0 ! A0 factors through v , we have that v k0 .A0 /  . h / k0 .A0 / D k0 .A0 /q ,
and so Proposition 7.17 shows that
                                                      v
                                         deg.A0 ! A0 /  q hdv :

As
                                    v        .7:16/                     .51/
                        deg.A0 ! A0 / D NE=Q . v / D NE=Q .phm
                                                            v
                                                               v
                                                                 /
we deduce that
                                           NE=Q .pm    dv
                                                  v /q :
                                                   v
                                                                                                        (52)
On taking the product over v, we find that
                                                          P
                                        NE=Q ./  q          vjp dv     qg :
   17 If E and k are number fields, then in general an isomorphism E ˝ k ! k n doesn’t induce an isomor-
                                                                          Q
phism OE ˝Z Ok ! Okn . Suppose, for example, that OE D ZŒX=.f .X// with f .X/ monic. Then the first
isomorphism exists if f .X/ splits in kŒX, but for the second to exist we need the roots to be distinct mod-
ulo every prime ideal in Ok . From a different perspective, Okn is étale over Ok , and so the existence of an
isomorphism would imply that OE is étale (i.e., unramified) over Z (by descent).
CHAPTER II. THE ARITHMETIC THEORY                                                                 68


But
                                      .7:16/                
                           NE=Q ./ D deg.A0 ! A0 / D q g ;
and so all the inequalities are equalities.
   Equality in (52) implies that
                                                      dv
                                NmE=Q .pm
                                        v / D Nm k=Q P
                                          v
                                                          ;

which equals
               Y                  Y
                                                NmE=Q .' 1 .Nm k='E P//
                                                                       
                          Nmk=Q P D
                   '2˚v                   '2˚v
                                               Y                     
                                      D Nm E=Q         ' 1 .Nm k='E P/ :
                                                            '2˚v

                                                            1 .Nm
                                              Q
From the definition of ˚v , we see that           '2˚v '            k='E P/ is a power of pv , and so
this shows that                    Y
                            pm
                             v D
                               v
                                                   ' 1 .Nmk='E P/:                              (53)
                                         '2˚v

On taking the product over v, we obtain the required formula.                                      2

C OROLLARY 8.2 With the hypotheses of the theorem, for all primes p of E dividing p,
                                        X
                       ordp ./ D               f .P='p/:                         (54)
                                         '2˚ , '     1 .P/Dp



Here 'p is the image of p in 'OE  'E  k.

P ROOF. Let p be prime ideal of OE dividing p, and let ' be the homomorphism E ! k.
If p D ' 1 .P/, then

                  ordp .' 1 .Nmk='E P// D ord'p Nmk='E P D f .P='p/;

and otherwise it is zero. Thus, (54) is nothing more than a restatement of (53).                   2

C OROLLARY 8.3 With the hypotheses of the theorem, for all primes v of E dividing p,

                                     ordv ./ j˚ \ Hv j
                                              D                                                 (55)
                                     ordv .q/   jHv j

where Hv D fW E ! k j  1 .P/ D pv g and q D .Ok W P/.

P ROOF. Because p is unramified in E, ordv .p/ D 1 for the primes v of E dividing p, and
so
                                 ordv .q/ D f .P=p/:
On the other hand (see (54)),
                                                  X
                                ordv ./ D                 f .P='pv /;
                                              '2˚\Hv

and so
                     ordv ./       X             1                    1
                              D                         D j˚ \ Hv j        :
                     ordv .q/                 f .pv =p/               jHv j
                                  '2˚\Hv                                                           2
CHAPTER II. THE ARITHMETIC THEORY                                                        69


R EMARK 8.4 The argument in the proof of (8.3) shows that (55) is equivalent to (54), even
without assuming that p is unramified in E (without that assumption, both of ordv .q/ and
jHv j have to be multiplied by e.pv =p/).

   We let ŒM R denote the class of an R-module M of finite length in the Grothendieck
group of such modules. For example, if R is a product of Dedekind domains, then every
R-module of finite length M has a composition series

       M      Mi  Mi 1      0; Mi =Mi 1  R=pi ; pi maximal ideal,

and the map M 7! i pi 1 defines an isomorphism of the Grothendieck group with the
                     Q
group of fractional ideals.

C OROLLARY 8.5 With the hypotheses of the theorem,

                                    ŒTgt0 .A0 /OE D ŒOE =OE OE :                    (56)

P ROOF. Let  D Ok =P. Then
                                                         M
                                          Tgt0 .A0 / '             ' ;
                                                             '2˚

where ' is a one-dimensional -vector space on which OE acts through '.18 If ' 1 .P/ D
p, then '  .OE ='p/q=.OE W'p/ , and so ' contributes pq=.OE W'p/ D pf .P=p/=f .'p=p/ D
pf .P='p/ to ŒTgt0 .A0 /OE . Thus, (56) is a restatement of (54).                      2

R EMARK 8.6 (a) In the statement of Theorem 8.1, k can be a number field.
    (b) The conditions in the statement are unnecesarily strong. For example, the formula
holds without the assumption that p be unramified in E. See Theorem 9.3 below.
    (c) When E is a subfield of k, Theorem 8.1 can be stated in terms of the reflex CM-type
cf. Shimura and Taniyama 1961, 13.

   Let .E; ˚/ be a CM-pair, and let E   Qal be the reflex field. Recall (1) that, for any
number field k, E   k  Qal , the reflex norm defines a homomorphism Nk;˚ from the
group of fractional ideals of k to that of E; if k contains all conjugates of E, then
                                         Y
                            Nk;˚ .a/ D            ' 1 .Nmk='E a/
                                                   '2˚

(see 1.26). Thus, the Shimura-Taniyama formula (50) says that

                                               ./ D Nk;˚ .P/:

Recall (1.23) that Nk;˚ .P/ D N˚ .Nmk=E  P/:

C OROLLARY 8.7 Let .E; ˚ / be a CM-pair, and let A be an abelian variety of CM-type
.E; ˚ / over a number field19 k which is a Galois extension of Q. Suppose that A has good
reduction at the prime ideal P of Ok . Let P \ OE  D p and p \ Z D .p/. Assume that (i)
p is unramified in E; (ii) End.A/ \ E D OE ; (iii) P is unramified over E  .
  18 With the notation of the proof of Theorem 8.1,  D k e and  D k .
                                                     '   0 '         0
  19 Thus, k contains the reflex field E  .
CHAPTER II. THE ARITHMETIC THEORY                                                           70


  (a) Let  be the Frobenius element .P; k=E  /; then there exists an a-multiplication
                                                                  .q/
      ˛W A ! A over a finite extension of k such that ˛0 W A0 ! A0 is the q-power Frobe-
      nius map, where q D .OE  W p/.
  (b) Moreover,
                                                      a D N˚ .p/:                          (57)

P ROOF. As  fixes E  , A and A have the same CM-type, and so they become isogenous
over a finite extension of k. Therefore, (a) holds by (7.42). Moreover,  f 1 ˛ ı  ı ˛ ı˛ D
, where f D Œ.P/W .p/; therefore,

                         af D N˚ .Nmk=E  P/ D N˚ .pf / D N˚ .p/f ;

which implies (57).                                                                          2

E XERCISE 8.8 Adapt the proof of Theorem 8.9 below to give a direct proof of Theorem
8.1 (with fewer assumptions).


    The original proof of the Shimura-Taniyama formula, which we presented above, is the
most direct and elementary. However, there other approaches which may add additional
insight for those with the appropriate knowledge.

Alternative approach using schemes (Giraud 1968)
Giraud (1968) re-examined the original proof from the perspective of schemes, and sharp-
ened some intermediate results, e.g., 7.29. He proved directly the formula for the tangent
space of an abelian variety with complex multiplication over a finite field, and deduced the
Shimura-Taniyama formula from it.
     In this subsection, A is an abelian variety of dimension g over a field k and R is a
                                                                def
commutative subring of End.A/ such that ŒRW Z D 2g and E D R ˝Z Q is a product of
fields. We make free use of the results from 7 on the functor M           AM from finitely
generated projective R-modules to abelian varieties.

T HEOREM 8.9 (G IRAUD 1968, T H ÉOR ÈM 1) Suppose k is finite, with q elements, and
that the q-power Frobenius map W A ! A lies in R. Then

                                         ŒTgt0 .A/R D ŒR=./R                            (58)

(equality of elements of the Grothendieck group of R).
                    T
P ROOF. Let ./ D i 2I qi be the primary decomposition of ./. Because nonzero         prime
                                                          20 Therefore, ./ D
                                                                              Q
ideals in R
          Q  are maximal,   the q i are relatively prime.                       i 2I qi , and
R=./ ' i 2I R=qi (Chinese remainder theorem).
    Write T .A/ for Tgt0 .A/. We can regard it as a commutative group scheme over k on
which R acts. For any finitely generated R-module M ,

                                             T .AM / ' T .A/M ;
  20 The only prime ideal containing q
                                         i is its radical pi . As pi ¤ pj , qi C qj D R.
CHAPTER II. THE ARITHMETIC THEORY                                                      71


as can be seen by using the definition of the tangent space in terms of dual numbers.21 As
T .A/ is killed by , we have T .A/ D T .A/R=./ , and so
                                                    def
                                    M
                           T .A/ D         Ti ; Ti D T .A/R=qi ;
                                                 i 2I

is the primary decomposition of T .A/. It suffices to prove that, for each i 2 I ,

                                             Card.Ti / D .RW qi /:                   (59)

In fact, since we know that
                                                        Y
                                 q g D Card.T .A// D     Card.Ti /
                                                            i 2I
                                             .7:16/        Y
                                 q g D deg./ D .RW .// D         .RW qi /;
                                                                              i 2I

it suffices to show that, for each i 2 I ,

                                             Card.Ti /  .RW qi /:                   (60)

As ./ is a free R-module, R=./ is of projective dimension  1. Therefore, each of the
direct factors of R=./ has projective dimension  1, and so qi is projective. Proposition
7.29 shows that deg.qi / D .RW qi /. On the other hand, M 7! T .A/M transforms the exact
sequence
                                   qi ! R ! R=qi ! 0
into the exact sequence
                                                                   T .qi /
                                   0 ! T .A/R=qi ! T .A/              ! T .A/qi ;

and so Proposition 7.17 shows that deg.qi /  Card.Ti /.                               2

P ROOF ( OF T HEOREM 8.1) We saw in the proof of (8.5) that, in the context of (8.1), for-
mulas (50) and (58) are equivalent.                                                     2


Alternative approach using p-divisible groups (Tate 1968)
Tate (1968, 5) restated the Shimura-Taniyama formula for p-divisible groups, and proved
it in that more general context. His proof also doesn’t require that p be unramified in E.
       21 For any k-algebra R,

                                        A.RŒ"/ ' A.R/ ˚ T .A/.R/              (*)
Therefore
                          HomR .M; A.RŒ"// ' HomR .M; A.R// ˚ HomR .M; T .A/.R//
i.e,
                                        AM .RŒ"/ ' AM .R/ ˚ T .A/M .R/.
On replacing A with AM in (*), we obtain

                                       AM .RŒ"/ ' AM .R/ ˚ T .AM /.R/:

On comparing these isomorphisms, we find that

                                               T .A/M ' T .AM /:
CHAPTER II. THE ARITHMETIC THEORY                                                       72


B RIEF REVIEW OF p- DIVISIBLE GROUPS
Define a p-divisible group G, including over a dvr.
Define its height, tangent space, and dimension.
Define the degree of an isogeny.
Note that Vp G is a Qp -vector space of dimension h.G/:

E XAMPLE 8.10 Let A be an abelian scheme, and let A.p/ D .Apn /n1 be the associated
p-divisible group. Then A A.p/ is a functor, and

                                    Tgt0 .A/ ' Tgt0 .A.p//:

The height of A.p/ is 2 dim A and the dimension of A.p/ is dim A.

   Need also:
  ˘ The degree of the q-power Frobenius map on a p-divisible group G over Fq is q d.G/ .
  ˘ The degree of qW G ! G on a p-divisible group G over Fq is q h.G/ :

p- DIVISIBLE GROUPS WITH COMPLEX MULTIPLICATION
Let k be a finite extension of Qp , and let G be a p-divisible group G over Ok . A subfield
E of End0 .G/ acts faithfully on Vp G, and so has degree  h.G/ over Qp . When equality
holds, we say that G has complex multiplication by E over Ok . Let kN be an algebraic
closure of k. Then                               M
                                Tgt.G/ ˝O kN '           kN' ;
                                                     '2˚

                         N and where kN' is a one-dimensional k-vector
where ˚  HomQp -alg .E; k/                                   N        space on which
                                         N
E acts through the homomorphism 'W E ! k. We say that G is of type .E; ˚ / over Ok .

P ROPOSITION 8.11 Let G be a p-divisible group with complex multiplication by E, and
let R D End.G/ \ E. For any ˛ 2 R, the degree of ˛W G ! G is .RW ˛R/:

P ROOF. To be added.                                                                     2

T HEOREM 8.12 (TATE 1968, T H ÉOR ÈME 3) Let G be a p-divisible group of type .E; ˚ /
over over Ok . Assume that R contains an element  which induces the q-power Frobenius
map on G0 over k0 . Then
                               ord./ d.G/         j˚j
                                       D        D       ;                         (61)
                                ord.q/     h.G/    jH j
                                                  N
where ord is the valuation on E and H D HomQp .E; k/:

P ROOF. We have
                           .8:11/                            0
               .Ok W Ok / D deg.G ! G/ D deg.G0 ! G0 / D q d.G/
                           .8:11/          q
               .Ok W qOk / D deg.G ! G/ D q h.G/ :

This proves the first equality, and the second is obvious.                               2
CHAPTER II. THE ARITHMETIC THEORY                                                                  73


P ROOF ( OF T HEOREM 8.1) We prove the equivalent statement (8.3). With the notation
of (8.1), let A be the abelian scheme over Ok with                  P let A.p/ be the
                                                 Q general fibre A, and
associated p-divisible group. Then E ˝Q Qp ' vjp E   Qv . Write 1 D    ev (in E ˝Q Qp /
with ev integral. Then x 7! ev x is an isogeny A.p/ ! vjp A.p/v . Let

                      Hv D fW E ! k j  1 .P/ D pv g ' fW Ev ! kg:

The idempotents ev also define a decomposition
                                                          M
                  Tgt0 .A/ ˝Q Qp ' Tgt0 .A.p// '                     Tgt0 .A.p/v /;
                                                               vjp

and A.p/v is of type .Ev ; ˚v /, where ˚v D ˚ \ Hv . Therefore, for G D A.p/v , equation
(61) becomes equation (55).                                                            2


Alternative approach using crystals (Deligne c1968)
In a handwritten manuscript (Deligne nd), Deligne showed how to derive the Shimura-
Taniyama formula, as well as the fundamental theorem of complex multiplication over the
reflex field, from the theory of canonical liftings of abelian varieties. See the next section.

Alternative approach using Hodge-Tate decompositions (Serre 1968)
Serre (1968) gives some hints for another possible approach to the formula of Shimura and
Taniyama in his remark on pages II-28 and II-29.
N OTES The first statement of the Shimura-Taniyama formula (50) in print is in Weil’s conference
talk (Weil 1956b, p. 21), where he writes:
      [For this] it is enough to determine the prime ideal decomposition of  . . . . But this
      has been done by Taniyama (cf. 3 of his contribution to this volume).
Italics in the original. In the mentioned section, Taniyama proves (50) by essentially the same
method as we used (Taniyama 1956).
     However, see the remark of Honda 1968, p. 89:
      [Theorem 8.1] was first mentioned in Taniyama 1956, where he reduced the proof to
      the case P was of absolute degree 1. In this case, the proof is direct (Shimura 1956,
      Taniyama 1956). But there was an error in this reduction of the proof and a complete
      proof was given by Shimura (cf. the footnote (2) of Taniyama 1957). According to his
      letter to the author, it is the following. . . .
    Taniyama 1957, footnote 2:
      A passage in the proof of proposition 3 in Taniyama 1956 (p. 38, l. 22) would indicate
      that N P0 D .P0 /  .P0 / determines the ideal decomposition of .P0 /, which is in
      fact not the case, since both pi and pi may divide P0 . It is easy however to amend this
      point and obtain the desired result. See the author’s forthcoming paper in collaboration
      with G. Shimura.
CHAPTER II. THE ARITHMETIC THEORY                                                            74


9     The fundamental theorem of complex multiplication over the
      reflex field.
Review of the reflex norm
Let E be a CM-algebra, and ˚ a CM-type on E, say, ˚  Hom.E; N          Q/, where N  Q is an
                                                                    N
algebraic closure of Q. By definition, E is the smallest subfield of Q such that there exists
an E ˝Q E  -module V with
                               M
                   V ˝E  N
                          Q'            N
                                        Q'      (as an E ˝Q NQ-module);
                                    '2˚

where QN ' is a one-dimensional Q-vector
                                   N        space on which E acts through '. The E ˝Q E  -
module V is then uniquely determined up to isomorphism. Let T D .Gm /E=Q and T  D
.Gm /E  =Q . Then there is a homomorphism N˚ W T  ! T of tori such that E  D T  .Q/ !
T .Q/ D E  sends a 2 E  to its determinant as an E-linear automorphism of V . From N˚
we get compatible homomorphisms on idèles, ideals, etc.. Moreover (1.26):
      Let k  Q  N be a finite extension of E  containing all conjugates of E. For any
      prime ideal a of k,
                                             Y
                          N˚ .Nmk=E  a/ D           ' 1 .Nmk='E a/:
                                                 '2˚


Preliminaries from algebraic number theory
L EMMA 9.1 Let a be a fractional ideal in E. For any integer m > 0, there exists an a 2 E 
such that aa  OE and .OE W aa/ is prime to m.
P ROOF. It suffices to find an a 2 E such that
                                    ordv .a/ C ordv .a/  0                                (62)
for all finite primes v, with equality holding if vjm.
    Choose a c 2 a. Then ordv .c 1 a/  0 for all finite v. For each v such that vjm or22
ordv .a/ < 0, choose an av 2 OE such that
                                 ordv .av / C ordv .c 1 a/ D 0
(exists by the Chinese remainder theorem). For any a 2 OE sufficiently close to each av
(which exists by the Chinese remainder theorem again), ca satisfies the required condition.2
     Let k be a number field. For a finite set S of finite primes of k, I S .k/ denotes the
group of fractional ideals of k generated by the prime ideals not in S . Assume k is totally
imaginary. Then a modulus m for k is just an ideal in Ok , and S.m/ denotes the set of
finite primes v dividing m (i.e., such that ordv .m/ > 0). Moreover, km;1 denotes the group
of a 2 k  such that
                                    ordv .a 1/  ordv .m/
for all finite primes v dividing m. In other words, a lies in km;1 if and only if multiplication
                                                                      ord .m/
by a preserves Ov  kv for all v dividing m and acts as 1 on Ov =pv v D Ov =m. Finally,
                                   Cm .k/ D I S.m/ = i.km;1 /
is the ray class group modulo m. Here i is the map sending an element to its principal ideal.
(Cf. CFT, V, 1).
    22 Or both!
CHAPTER II. THE ARITHMETIC THEORY                                                                           75


The fundamental theorem in terms of ideals
Let NQ be an algebraic closure of Q, and let A be an abelian variety with complex multi-
plication by a CM-algebra23 E over Q.   N Let ˚  Hom.E; Q/    N be the type of A. Assume
End.A/ \ E D OE . Fix an integer m > 0.
    Let  2 Gal.Q=E N    /. Because  fixes E  , the varieties A and A are E-isogenous

(3.12), and so there exists an a-multiplication ˛W A ! A for some ideal a  OE (see 7.31).
Recall (7.29) that ˛ has degree .OE W a/. After possibly replacing ˛ with ˛ ı a for some
a 2 a 1 , it will have degree prime to m (apply 9.1). Then ˛ maps Am isomorphically onto
Am .24
                Q                                          def Q
    Let Zm D `jm Z` and Om D OE ˝ Zm . Then Tm A D `jm T` A is a free Om -module
of rank 1 (see 7.6). The maps

                                  x 7! x, x 7! ˛xW Tm A ! Tm .A/

are both Om -linear isomorphisms, and so they differ by a homothety by an element ˇ of
Om:

                              ˛.ˇx/ D x; all x 2 Tm A:
For any b 2 OE sufficiently close to ˇ, ˛ ı b will agree with  on Am . Thus, after replacing
˛ with ˛ ı b, we will have

                                 ˛.x/  x       mod m;       all x 2 Tm A:

Now ˛ is an a-multiplication for an ideal a that is well-defined up to an element of i.Em;1 /.

R EMARK 9.2 The abelian variety A will have a model, which we again denote A, over
                   N that is finite and Galois over E  . After possibly enlarging k, we may
some subfield k of Q
suppose that A has complex multiplication by E over k. Let P be a prime ideal of k such
that
  ˘ A has good reduction at P;
                                    def
  ˘ P is unramified over p D P \ OE  ;
          def
  ˘ p D p \ Z is unramified in E.
Let  be the Frobenius element .P; k=E  /, and let q D .OE  W p/. Corollary 8.7 shows
                                                                     .q/
that there exists an a-multiplication ˛W A ! A such that ˛0 W A0 ! A0 is the q-power
Frobenius map; moreover, a D N˚ .p/. For any m prime to p and such that Am .k/ D
Am . N
     Q/, the homomorphism ˛ agrees with  on Am .k/ (because it does on A0;m .k0 / '
Am .k/).

    Let  0 be a second element of Gal.Qal =E  /, and let ˛ 0 W A !  0 A be an a0 -multiplication
acting as  0 on Am (which implies that its degree is prime to m). Then  ˛ 0 is again an a0 -
multiplication (obvious from the definition 7.19), and so  ˛ 0 ı ˛ is an aa0 -multiplication
A !  0 A (see 7.28) acting as  0  on Am . Therefore, we have a homomorphism  7!
a. /W Gal.Qal =E  / ! Cm .E/. This homomorphism factors through Gal.k=E  / for some
  23 Is it in fact necessary to assume that E is a CM-algebra in all of this? Later we need a polarization whose

Rosati involution is complex conjugation on E, which implies that E is a CM-algebra.
  24 Here A D Ker.A m                                               N which can be identified with Am .Q/.
                            ! A/. It is an étale group scheme over Q;                                    N
              m
CHAPTER II. THE ARITHMETIC THEORY                                                            76


finite abelian extension k of E  , which we may take to be the ray class field for a modulus
m of E  . Thus, we obtain a well-defined homomorphism

                                       Cm .E  / ! Cm .E/

sending an ideal a in I S.m/ .E  / to the ideal of E associated with  D .a ; k=E  /.

T HEOREM 9.3 The above homomorphism is that defined by N˚ .

P ROOF. It suffices to verify this for a set of generators for Cm .E  /. Let k  N
                                                                                  Q be a field
                                                             
finite and Galois over E , containing the ray class field Em , and such that A has a model
over k with complex multiplication by E over k for which Am .k/ D Am . N        Q/. Then 9.2
shows that the two homomorphisms agree on the primes P \ OEm , where P is a prime of k
satisfying the conditions of 9.2. By Dirichlet’s theorem on primes in arithmetic progressions
(CFT, V, 2.5), the classes of these primes exhaust Cm .                                      2


More preliminaries from algebraic number theory
We write art for the reciprocal reciprocity map, i.e., artk .s/ D reck .s/ 1 for s 2 Ak
                                                                                         . When
k is totally imaginary, it factors through A
                                            f;k
                                                , and we  also  write art k for the map A  
                                                                                           f;k
                                                                                               !
       ab
Gal.k =k/ that it defines; then

                                    artk W A          ab
                                            f;k ! Gal.k =k/

is surjective with kernel the closure of k  (embedded diagonally) in A
                                                                       f;k
                                                                           .
    Let cyc W Gal.Qal =Q/ ! b Z be the cyclotomic character:

                                             D  cyc . /

for all roots  of 1 in C.

L EMMA 9.4 For any  2 Gal.Qal =Q/,

                                     artQ .cyc . // D jQab :

P ROOF. Exercise.                                                                             2

L EMMA 9.5 Let E be a CM-field. For any s 2 A
                                             f;E
                                                 ,

                                NmE=Q .s/ 2 cyc .artk .s//  Q>0 :

P ROOF. Let  2 Gal.Qal =E/ be such that artE .s/ D  jE ab . Then

                                    artQ .NmE=Q .s// D  jQab

by class field theory, and so

                                artQ .NmE=Q .s// D artQ .cyc . //:

The kernel of artQ W A
                      f
                        ! Gal.Qab =Q/ is A
                                          f
                                            \.Q  R>0 / D Q>0 (embedded diagonally).2

L EMMA 9.6 For any CM-field E, the kernel of artE W A    f;E
                                                              =E  ! Gal.E ab =E/ is uniquely
divisible by all integers, and its elements are fixed by E .
CHAPTER II. THE ARITHMETIC THEORY                                                                  77


P ROOF. The kernel of artE is E  =E  , where E  is the closure of E  in A    f;E
                                                                                       . It is also
equal to UN =U for any subgroup U of OE of finite index. A theorem of Chevalley (see
                                            

Serre 1964, 3.5) shows that A f;E
                                   induces the pro-finite topology on U . If we take U to be
contained in the real subfield of E and torsion-free, then it is clear that UN =U is fixed by E
and (being isomorphic to .b Z=Z/ŒE WQ=2 ) uniquely divisible.                                    2

L EMMA 9.7 Let E be a CM-field and let ˚ be a CM-type on E. For any s 2 A
                                                                         f;E 
                                                                               ,

                           N˚ .s/  E N˚ .s/ 2 cyc .artE  .s//  Q>0 :

P ROOF. According to (10),

                                N˚ .s/  E N˚ .s/ D NmE  =Q .s/;

and so we can apply (9.5).                                                                          2

L EMMA 9.8 Let E be a CM-field and ˚ a CM-type on E. There exists a unique homo-
morphism Gal.E ab =E  / ! Gal.E ab =E/ rendering

                                                  N˚
                                   A
                                    f;E 
                                                              A
                                                               f;E

                                       artE                      artE

                              Gal.E ab =E  /           Gal.E ab =E/

commutative.

P ROOF. As artE W A f;E
                          ! Gal.E ab =E/ is surjective, the uniqueness is obvious. On the
other hand, N˚ maps E  into E  and is continuous, and so it maps the closure of E 
into the closure of E  .                                                               2

P ROPOSITION 9.9 Let s; s 0 2 A
                               f;E 
                                     . If artE  .s/ D artE  .s 0 /, then N˚ .s 0 / 2 N˚ .s/  E  .

P ROOF. Let  2 Gal.Qal =Q/ be such that

                                  jE ab D artE  .s/ D artE  .s 0 /.

Then (see 9.5),

                   N˚ .s/  E N˚ .s/ D cyc . /  E  D N˚ .s 0 /  E N˚ .s 0 /.

Let t D N˚ .s/=N˚ .s 0 / 2 A  f;E
                                   . Then t 2 Ker.artE / and t  E t 2 E  . As the map x 7!
x  E x is bijective on Ker.artE /=E  (see 9.6), this shows that t 2 E  .                2


The fundamental theorem in terms of idèles
Let N
    Q be an algebraic closure of Q, and let A be an abelian variety with complex multipli-
cation by a CM-algebra E over Q.N Let ˚  Hom.E; Q/  N be the type of A.
                 N                          
    Let  2 Gal.Q=E /. Because  fixes E , there exists an E-isogeny ˛W A ! A (see
3.12). The maps
                          x 7! x, x 7! ˛xW Vf .A/ ! Vf .A/
CHAPTER II. THE ARITHMETIC THEORY                                                                      78

                 def
are both Af;E D E ˝ Af -linear isomorphisms. As Vf .A/ is free of rank one over Af;E ,25
they differ by a homothety by an element . / of A
                                                   f;E
                                                       :

                                 ˛.. /x/ D x;       all x 2 Vf .A/:                               (63)

When the choice of ˛ is changed, . / is changed only by an element of E  , and so we
have a well-defined map
                            W Gal.Qal =E  / ! A      
                                                 f;E =E :                           (64)
The content of the next theorem, is that . / equals N˚ .s/ mod E  for any s 2 A
                                                                                  f;E 
                     ab
with artE  .s/ D jE .

T HEOREM 9.10 Let A be an abelian variety with complex multiplication by a CM-algebra
       N and let  2 Gal.Q=E
E over Q,                N    /. For any s 2 A     with artE  .s/ D jE ab , there is a
                                               f;E 
unique E-“isogeny” ˛W A ! A such that ˛.N˚ .s/  x/ D x for all x 2 Vf A:

R EMARK 9.11 (a) It is obvious that ˛ is determined uniquely by the choice of s 2 A             f;E
such that rec.s/ D jE ab . If s is replaced by s 0 , then N˚ .s 0 / D a  N˚ .s/ with a 2 E  (see
9.9), and ˛ must be replaced by ˛  a 1 .
    (b) The theorem is a statement about the E-“isogeny” class of A — if ˇW A ! B is an
E-“isogeny”, and ˛ satisfies the conditions of the theorem for A, then ˇ ı ˛ ı ˇ 1 satisfies
the conditions for B:
    (c) Let ˛ as in the theorem, let  be a polarization of A whose Rosati involution induces
E on E, and let W Vf AVf A ! Af .1/ be the Riemann form of . Then, for x; y 2 Vf A;
                                         def
                        . /.x; y/ D  . .x; y// D cyc . /  .x; y/

because     .x; y/ 2 Af .1/. Thus if ˛ is as in the theorem, then

  cyc ./  .x; y/ D . /.N˚ .s/˛.x/; N˚ .s/˛.y// D . /.N˚ .s/N˚ .s/˛.x/; ˛.y//

and so
                                   .c /.x; y/ D . /.˛x; ˛y/;
with c D cyc ./=NE  =Q .s/ 2 Q (see 9.5).
             
   Let T E and T E be the algebraic tori over Q with Q-points E  and E  respectively.
The norm a 7! a  E a defines a homomorphism T E ! T F , and we define T to be the fibre
product T D Gm T F T E :
                                               T         TE
                                                                                                     (65)

                                           Gm            TF:
We begin with two easy lemmas.
  25 Let R D End.A/ \ E. For all `, V A is free of rank one over E def
                                         `                        ` D E ˝Q Q` , and for all ` not dividing
                                           def
.OE W R/, T` A is free of rank one over R` D R ˝Z Z` (see 7.6).
CHAPTER II. THE ARITHMETIC THEORY                                                             79


L EMMA 9.12 The map
                       T .Af /=T .Q/ ! T E .Af /=T E .Q/ D A
                                                            f;E =E
                                                                   


defined by the homomorphism T ! T E is injective, and realizes the first group as a topo-
logical subspace of the second.
P ROOF. Let a 2 T .Af / map to b 2 T E .Q/. To lie in T .Q/, b must satisfy a polynomial
equation. Because it satisfies this equation in Af , it satisfies it in Q  Af .
    That T .Af /=T .Q/ is a topological subspace of T E .Af /=T E .Q/ is obvious: T .Af /
is certainly a topological subspace of T E .Af /, and T .Af /=T .Q/ and T E .Af /=T E .Q/
are both endowed with the quotient topology.                                            2

L EMMA 9.13 The space T .Af /=T .Q/ is Hausdorff.
                                                                          
P ROOF. Q In fact, T .Q/ is even discrete in T .Af /. Note that T .Q/ \ OE  , being equal to
                                                                        
T .Q/ \ v finite OE;v , is open in T .Q/. Moreover, it contains T .Q/ \ OF as a subgroup of
finite index. But
                             T .Q/ \ OF
                                          D fa 2 OF
                                                      j a2 D ˙1g
                            
is finite, and so T .Q/ \ OE  is finite and open in T .Q/. It follows easy that easily that T .Q/
is discrete in T .Af /. In particular, it is closed (Hewitt and Ross 1963, II 5.10).            2

   Thus, elements a; b of T .Af /=T .Q/ are equal if a 2 bU for all open neighbourhoods
U of 1 in A
           f;E
               =E  .26

L EMMA 9.14 Let  2 Gal.Qal =Q/. Let s be an element of A    f;E 
                                                                    such that artE  .s/ D
jE ab , and let ./ be an element of A
                                         f;E
                                             such that (63) holds for some E-“isogeny”
˛W A ! A. Then . /=N˚ .s/ mod E  lies in the subgroup T .Af /=T .Q/ of A   f;E
                                                                                     =E  .
P ROOF. We know that
                                        .1:24/                     .9:5/
                   N˚ .s/  E N˚ .s/ D NmAf;E  =Af .s/ D cyc . /  a
for some a 2 Q>0 .
    A calculation as in (9.11c) shows that,
                         .c /.x; y/ D . /.˛x; ˛y/; all x; y 2 A
                                                                f;E ;                       (66)
with c D cyc ./= .. /  E . //. But it follows from the last paragraph of (2.9) that (66)
holds with c a totally positive element of F . Thus
                              . /  . / D cyc . /=c;     c 2 F0 ;                    (67)
where ./ D E . /.
   Let t D ./=N˚ .s/. Then
                                         t tN D 1=ac 2 F0 :                                (68)
Being a totally positive element of F , ac is a local norm from E at the infinite primes, and
(68) shows that it is also a local norm at the finite primes. Therefore we can write ac D e eN
for some e 2 E  . Then
                                          t e  t e D 1:
Thus, te 2 T .Af /, and it represents t  E  in A
                                                  f;E
                                                      =E  .                                   2
  26 Because then b 1 a lies in all open neighbourhoods of 1 in T .A /=T .Q/:
                                                                    f
CHAPTER II. THE ARITHMETIC THEORY                                                           80


     Note that (1.24) shows that N˚ .s/ 2 T .Af /, and so the map W Gal.Qal =E  / ! A f;E
                                                                                             =E 
in (64) takes values in the Hausdorff subgroup T .Af /=T .Q/ of A    f;E
                                                                          =E  . It is a homo-
morphism27 , and so it factors through Gal.Qal =E  /ab . When combined with the Artin map,
it gives a homomorphism 0 W A  f;E 
                                       =E  ! Af;E
                                                     =E  .
     In order to prove Theorem 9.10, it remains to show N˚ .s/ 2 0 .s/U for all open neigh-
bourhoods U of 1 in A   f;E
                             =E  . Choose an integer m > 0. For some modulus m, there
exists a commutative diagram

                                    A
                                     f;E 
                                           =E                   A
                                                                   f;E
                                                                       =E 


                                       Cm .E  /                       Cm .E/

in which the vertical maps are given by class field theory (CFT, V, 4.6) and the horizontal
maps are given either by N˚ or by 0 . On the bottom row, these maps agree by Theorem
9.3, which implies that they agree on the top row, because the kernels of the homomor-
phisms Af;E
             =E  ! Cm .E/, as m runs over the positive integers, form a basis for the open
neighbourhoods of 1 in Af;E
                             =E  (cf. CFT, V, 4.6).

R EMARK 9.15 Theorem 9.10 holds with N Q replaced by C, and  taken to be any automor-
                   
phism of C fixing E . This follows immediately from the theorem because of (7.10).

E XERCISE 9.16 Rearrange the proof by first showing that the two homomorphisms

                                      Gal.E ab =E  / ! A    N
                                                          f;E =E

agree; then use the polarization to obtain the stronger result.

The fundamental theorem in terms of uniformizations
Let .A; i W E ,! End0 .A// be an abelian variety with complex multiplication over C, and let
 be a polarization of .A; i /. Recall (3.11, 3.17) that the choice of a basis element e0 for
H0 .A; Q/ determines a uniformization W C˚ ! A.C/, and hence a quadruple .E; ˚ I a; t /,
called the type of .A; i; / relative to .

T HEOREM 9.17 Let .A; i; / be of type .E; ˚ I a;t / relative to a uniformization  W C˚ !
A.C/, and let  be an automorphism of C fixing E  . For any s 2 A
                                                                  f;E 
                                                                        such that artE  .s/ D
                                                             0
jE ab , there is a unique uniformization  0 W C˚ ! .A/.C/ of A such that
   27 Choose E-isogenies ˛W A ! A and ˛ 0 W A !  0 A0 , and let


                                                  ˛.sx/ D x
                                                ˛ 0 .s 0 x/ D  0 x:

Then ˛ 0 ı ˛ is an isogeny A !  0 A, and

                                      .˛ 0 ı ˛/.ss 0 x/ D .˛ 0 /.˛.ss 0 x//
                                                        D .˛ 0 /..s 0 x//
                                                        D .˛ 0 .s 0 x//
                                                        D  0 x:
CHAPTER II. THE ARITHMETIC THEORY                                                         81


  (a) .A; i; / has type .E; ˚ I f a; t  cyc . /=f fN/, where f D N˚ .s/ 2 A
                                                                               f;E
                                                                                   I
  (b) the diagram
                                                        0
                                              E=a                A.C/
                                                                   
                                                        00
                                          E=f a               A.C/
      commutes, where 0 .x/ D ..'x/'2˚ / and 00 .x/ D  0 ..'x/'2˚ 0 /:

P ROOF. According to Theorem 9.10, there exists an isogeny ˛W A ! A such that ˛.N˚ .s/
x/ D x for all x 2 Vf A. Then H1 .˛/ is an E-linear isomorphism H1 .A; Q/ ! H1 .A; Q/,
and we let  0 be the uniformization defined by the basis element H1 .˛/.e0 / for H1 .A; Q/.
The statement now follows immediately from Theorem 9.10 and (9.11c).                       2


The fundamental theorem in terms of moduli
R EVIEW OF THE SETTING
Recall that S is the real torus with S.R/ D C . There are characters z and zN of S inducing
the maps z 7! z and z 7! zN respectively on the real points of S,

                             C D S.R/  S.C/ ⇒ Gm .C/ D C :

Let  be the cocharacter of S such that
                                  
                                     z ı  D idGm
                                                  :
                                     zN ı  D 1

The characters z; zN of S define an isomorphism
                                              .z; Nz/
                                        SC        ! Gm  Gm                             (69)

and  is the cocharacter of SC such that .x/ maps to .x; 1/ in Gm  Gm .
   Let .E; ˚/ be a CM-pair,    and let T E D .Gm /E=Q . As noted in 1, ˚ defines an iso-
morphism E ˝Q R ! '2˚ C, and hence an isomorphism28
                      Q


                                               TRE ' S˚ :                               (70)

Define
                                              h˚ W S ! TRE
to be the homomorphism whose composite with (70) is

                                          z 7! .z; : : : ; z/:
                                Q
The isomorphism E ˝Q C '            '2I C, I D Hom.E; C/, defines an isomorphism

                                          TCE ' .Gm /I :

The cocharacter
                                        def
                                    ˚ D h˚ ı W GmC ! TCE
  28 By S˚ we mean a product of copies of S indexed by ˚.
CHAPTER II. THE ARITHMETIC THEORY                                                           82


corresponding to h˚ satisfies
                                              
                                                  z if ' 2 ˚
                                 ˚ .z/' D
                                                  1 if ' … ˚:
    Recall that the reflex field E  of .E; ˚ / is the subfield of C generated by the elements
                                    X
                                             '.a/; a 2 E:
                                        '2˚

It can also be described as the field of definition of ˚ :
     We are interested in abelian varieties A of CM-type .E; ˚ / over fields k containing E  .
Recall that this means that there is given a homomorphism i W E ! End0 .A/ such that
                                               X
                         Tr.i.a/j Tgt0 .A// D           '.a/, all a 2 E;                  (71)
                                                    '2˚

(equality of elements of k).

S TATEMENT AND PROOF
Let .E; ˚/ be a CM-pair, and let T D Ker.T E ! T F =Gm / as in (65). Let V be a one-
dimensional E-vector space. Note that T E acts on V , and .V; h˚ .i // is the rational Rie-
mann pair attached to .E; ˚ / (cf. 2.5). Let be a rational Riemann form on .V; h˚ .i //
(cf. 2.9). Thus, is an alternating form V  V ! Q such that .x; y/ 7! .x; h˚ .i /y/ is a
positive definite symmetric form on V ˝ R. Note that T is the subtorus of GL.V / such that
   T .R/ D f˛ 2 GLR˝Q E .V / j 9.˛/ 2 R such that        .˛x; ˛y/ D .˛/ .x; y/g        (72)
for all Q-algebras R.
P ROPOSITION 9.18 For any compact open subgroup K of T .Af /, the set T .Q/nT .Af /=K
classifies the isomorphism classes of quadruples .A; j; ; K/ in which
  ˘ A is a complex abelian variety,
  ˘  is a polarization of A,
  ˘ j is a homomorphism E ! End0 .A/, and
  ˘ K is a K-orbit of E-linear isomorphisms W V .Af / ! Vf .A/ sending to an A        f
                                                                                           -
       multiple of 
satisfying the following condition:
       (*) there exists an E-linear isomorphism aW H1 .A; Q/ ! V sending  to a
       Q -multiple of .
An isomorphism from one quadruple .A; j; ; K/ to a second .A0 ; j 0 ; 0 ; 0 K/ is an E-
“isogeny” sending  to a Q -multiple of 0 and  to 0 modulo K.
P ROOF. Choose an isomorphism aW H1 .A; Q/ ! V as in (*), and consider29
                                                     a
                                V .Af / ! Vf .A/ ! V .Af /.
Then a ı  satisfies (72) with R D Af , and so a ı  2 T .Af /. The isomorphism a is deter-
mined up to composition with an element of T .Q/, and  is determined up to composition
with an element of K. Therefore, the class of a ı  in T .Q/nT .Af /=K is well-defined. It
remains to show that the map .A; j; ; K/ 7! Œa ı  is surjective and that its fibres are the
isomorphism classes, but this is routine.                                                     2
  29 Recall that V A ' H .A; Q/ ˝ A .
                  f     1          f
CHAPTER II. THE ARITHMETIC THEORY                                                        83


T HEOREM 9.19 Let  be an automorphism of C fixing E  . If .A; j; ; K/ satisfies (*),
then so also does .A; j; ; K/. Moreover, the isomorphism class of  .A; j; ; K/ de-
pends only on jE ab . For any s 2 A
                                     f;E 
                                           such that artE  .s/ D  jE ab ,
                   .A; j; ; K/  .A; j; ; f K/; where f D N˚ .s/.
P ROOF. This follows immediately from (9.10) and (9.11c).                                 2

R EMARK 9.20 Let M be the set of isomorphism classes quadruples satisfying (*). Propo-
sition 9.18 says that
                             M ' T .Q/nT .Af /=K.
Theorem 9.19 says that this isomorphism is equivariant for the following action of Aut.C=E  /
on the right hand side: for  2 Aut.C=E  /, choose an s 2 A    f;E 
                                                                       such that artE  .s/ D
    ab
jE ; then, for a 2 T .Af /,  Œa D ŒN˚ .s/  a.
R EMARK 9.21 In both Proposition 9.18 and Theorem 9.19, it is possible to replace C with
Qal (apply 7.10).

Alternative approach using crystals (Deligne c1968)
In a handwritten manuscript (Deligne nd), Deligne showed how to derive the Shimura-
Taniyama formula, as well as the fundamental theorem over the reflex field, from the theory
of canonical liftings of abelian varieties. The remainder of this section is based on his
manuscript. [There may be some sign problems here.]

R EVIEW OF THE CRYSTALS ATTACHED TO AN ABELIAN SCHEME
Let A be an abelian scheme of relative dimension g over W .Fq /, the ring of Witt vectors
with entries in Fq . To avoid possible problems, we assume p ¤ 2. We are interested in the
crystalline H1 of A (alias, Dieudonné module).
                      def
9.22 Attached to A0 D A mod p, there are the following objects.
 (a) A free W .Fq /-module M D M.A0 / of rank 2g (the crystalline H1 of A0 , alias the
     covariant Dieudonné module).
  (b) Let  be the automorphism of W .Fq / lifting the q-power Frobenius automorphism
      on Fq , and consider the standard diagram
                                        Frob      .p/
                                A0              A0                  A0


                                                         x7!x q
                                               Spec Fq            Spec Fq :

      Then
                                  .p/                               def       1
                             M.A0 / D M ˝W .Fq /; W .Fq / D M                   :
                      1
      Note that M        can be identified with M but with w 2 W .Fq / acting according to
      the rule
                                         w  m D  1 .w/  m:
      Because M is a covariant functor, there is a W .Fq /-linear map
                                                                     1
                                     F D M.Frob/W M ! M                  :
CHAPTER II. THE ARITHMETIC THEORY                                                                          84


  (c) Moreover,
                                                 1
                                                         _ def
                         M=pM D M ˝W .Fq / Fq ' HdR .A0 / D H1dR .A0 /,

       and hence there is a filtration on M=pM
                                                        1
                                      M=pM D F               F 0  F 1 D 0;
                                        1                             1 =F 0 D Tgt .A/, and so
       dual to the Hodge filtration on HdR .A0 /. Here F                          0

                                                                            1
                                       F .M=pM /  F 0 .M=pM /                 :                        (73)
                                                            1
  (d) There exists a W .Fq /-linear map V W M                  ! M such that F V D VF D p (because
      we are considering H1 ).

9.23 Attached to A, there is an isomorphism of M with the de Rham homology of A=W .Fq /
                                                 1
                                                                 _
                                            M ' HdR .A/

                                                    1
                                                           _
which is compatible with the isomorphism M=pM ' HdR   .A0 / in (9.22c). In particular,
there is a filtration on M ,
                               M D F 1  F 0  F 1 D 0;
                                 1
dual to the Hodge filtration on HdR .A/.

A PPLICATION TO ABELIAN VARIETIES WITH COMPLEX MULTIPLICATION
Let A be an abelian scheme over W .Fq /, as in the last subsubsection. Assume that the
general fibre A1 of A over B.Fq / has complex multiplication by the CM-algebra E, and let
˚ be its CM-type. Thus
                                          X
                        Tr.aj Tgt0 .A// D        '.a/, all a 2 E;                   (74)
                                                       '2˚

(equality of elements of B.Fq /), and E  is the subfield of B.Fq / generated by these ele-
ments.30
    Let .M; F / be as in (9.23). The algebra E acts on
                                      def
                              M Œ p1  D M ˝W .Fq / B.Fq / ' H1dR .A1 /:

Over E  , and a fortiori over B.Fq /, we have a homomorphism ˚ W Gm ! T E , and hence
a homomorphism
                               ˚ W B.Fq / ! .E ˝Q B.Fq // :
The module M Œ p1  is free of rank one over E ˝Q B.Fq /. It follows from (74) that there is a
decomposition
                               M Œ p1  D M Œ p1  1;0 ˚ M Œ p1 0; 1                    (75)
   30 Note that we get this situation when we start with an abelian variety A with complex multiplication by E
                                                                           def
over a number field k and a prime ideal P in k that is unramified over .p/ D Z \ P and at which A has good
reduction.
CHAPTER II. THE ARITHMETIC THEORY                                                                          85


such that
                           ˚ .x/ acts on M r; s as multiplication by x r ;
                       
                           M 0; 1 D F 0 .M Œ p1 /:

    Let OA D End.A/ \ E, and let OpA D OA ˝Z Zp . If ˚ .p/ 2 OpA ˝ W .Fq /, for exam-
ple, if OA is maximal at p, then (75) gives31

                                     1;0                        def
                         M DM              ˚ M 0; 1 ;    M r;s D M \ M Œ p1 r;s :                       (76)

P ROPOSITION 9.24 Suppose that
  (a) ˚ .p/ 2 OpA ˝ W .Fq /
 (b) the residue field of E   B.Fq / is Fp .
Then A0 is ordinary, and A is the canonical lifting of As

P ROOF. From (b) we have that

                                         E  D B.Fp /  B.Fq /;

and so ˚ defines a homomorphism

                                   ˚ W B.Fp / ! .E ˝Q B.Fp // :

The decomposition (76), which exists because of (a), is compatible with F because the
action of E (contrary to that of E ˝Q B.Fq /) commutes with F . From (73), we find that

                                               1;0              1;0       1
                                       F .M          /  p.M          /
                                     F n .M    1;0
                                                     /  q.M    1;0
                                                                      /:

We have written q D p n , so that F n is the q-power Frobenius endomorphism of A0 . There-
fore, the p-adic valuations of half of the eigenvalues of the Frobenius are  n, which implies
A0 is ordinary, and          (                                   1
                                F .M 1;0 / D p.M 1;0 /
                                                               1                         (77)
                                F .M 0; 1 / D .M 0; 1 / :
   The decomposition (76) corresponds to a decomposition of Tp .A/, which, by (77),
shows that A is the canonical lifting of A0 .                                    2

    One pulls also from (4), under the hypothesis (a,b) of Proposition 1

C OROLLARY 9.25 Under the hypotheses (a,b) of Proposition 9.24, the canonical lifting of
A.p/ is the abelian variety p-isogenous to A with the “Dieudonné module”

                                      M.A.p/ / D ˚ .p/  M.A/:

P ROOF. Follows from (77).                                                                                   2

R EMARK 9.26 The converse of Proposition 9.24 is true.
   31 Let x 2 M Œ 1  decompose into x D x C x with x 2 M Œ 1 0; 1 and x 2 M Œ 1  1;0 , and set ˛ D .p/.
                  p                          0     1       0       p          1       p
Then ˛ r x D x0 Cp r x1 , and so, if x 2 M , then ˛ r x ! x0 as r ! 1 which implies that x0 2 M \M Œ p1 0; 1 D
M 0; 1 . Therefore also x1 D x x0 2 M \ M Œ p1  1;0 D M 1;0 .
CHAPTER II. THE ARITHMETIC THEORY                                                            86


M ODULI
We now let .E; ˚/ be a CM-pair with ˚  Hom.E; C/ (so that now E   C). Let k be a
finite Galois extension of E  with Galois group G. Let V be a free E-module of rank 1,
and VZ be a lattice in V stable under OE . Let K be a compact open subgroup of A
                                                                                E;f
                                                                                    that
          def
        Z D VZ ˝ Z invariant.
leaves Vb          b
    When B is an abelian variety, we write B ˝ Q for the abelian variety up to isogeny
underlying it; put
                                           Y
                                   b.B/ D
                                   T           T` .B/;
                                                        `
                                   b.B ˝ Q/ D T
                                   V          b.B/ ˝Z Q:

Note that B is determined by the pair .B ˝ Q; T b.B/  V b.B ˝ Q//.
   Let K M.k/, or simply M , be the set of geometric isomorphism classes (˛  ˇ if ˛ is
isomorphic to ˇ after an extension of scalars) of objects .A; ; N/ consisting of
  ˘ an abelian variety up to isogeny A=k, with complex multiplication by E satisfying
      (71);
  ˘ a polarization  of A (i.e., an isogeny W A ! A_ defined by an ample line bundle);
  ˘ a class mod K, N, of E-linear isomorphisms.

                                          W V ˝Q Af ! V
                                                       b.A/

         Thus, for any  2 N, N D K. We require the class to be defined over k, not its
         elements.
       Denote by AN the abelian variety endowed with an isomorphism AN ˝ Q ! A ˝ Q such
that
                                          b.AN / D .Vb/
                                          T            Z
(the right hand side is independent of  2 N). It has complex multiplication by OE .
    The group A                                        
                 E acts on M (through its quotient AE;f ) according to the rule:

                            .A; ; N/  a D .A; ; Na/;    a 2 A
                                                                  E;f :

    Let F  E be a product of the largest totally real subfields of the factors of E. Then
c 2 F  acts on M according to the rule:

                                    c  .A; ; N/ D .A; c; c N/;

where
                                 c D  ı c D c  ı W A ! A_ :
    The Galois group G acts on M and commutes with the actions of A          E;f
                                                                                   and F  . The
                             
commutative group F  AE;f acts transitively on M , and so G acts through its largest
abelian quotient G ab .
    Denote by ŒA; ; N 2 M the geometric isomorphism class of .A; ; N/. The set of primes
of E  that are unramified and have degree 1 (i.e., have residue field the prime field) has
density 1. In order to calculate the action of G ab on M , it suffices therefore to calculate

                        v .ŒA; ; N/      .v the Frobenius map at v)

for the primes v of E  such that
CHAPTER II. THE ARITHMETIC THEORY                                                             87


  (a) Ev D Qp (p the residue characteristic at v);
  (b) p ¤ 2 and v is unramified in E;
  (c) K  .OE;p / ;
  (d) AN has good reduction at the primes of k above the v.
We now fix such a v.
    Since we are interested only in the geometric isomorphism classes, we may, when cal-
culating v .ŒA; ; N/ extend scalars from k to kv0 for v 0 jv. We are then in the situation of
Proposition 9.24. Thanks to (c), to give N amounts to giving
  ˘ AN and
                                                     N
  ˘ an isomorphism class N n W VZ =nVZ ! An (mod K) for any sufficiently large n prime
        to p:
    After Proposition 9.24, v .ŒA; ; N/ is defined by the isomorphism class of the canoni-
                                                             .p/
cal lifting of .A0 ; 0 ; N/.p/ . The canonical lifting of A0 is A˚ .p/N , and so

                                  v .A; ; N/  .A; p; N˚ .p//:                        (78)

    Consider now the set MN of geometric isomorphism classes of objects .A; ;N N/ as above,
                N
except that now  is a homogeneous polarization (i.e., given up to a factor in Q ).
                                 
    The homomorphism N˚ W T E ! T E is that deduced from ˚ W GmE  ! TEE by taking
the norm:
                                ResE  =Q .˚ /                    Norm
                        TE                   ! ResE  =Q .TEE /        ! T E:
Let 'W A
        E  ! Gal.E
                    al =E  /ab be the reciprocity homomorphism of global class field the-

ory.

P ROPOSITION 9.27 The action of Gal.E al =E  /ab on MN is given by the following rule:
let e 2 A                               
         E  have finite component ef 2 AE  ;f ; then

                                          N N/ D ŒA; ;
                                  '.e/ŒA; ;          N NN˚ .ef /:

P ROOF. This formula is compatible with (78), and so it suffices to check that it defines an
action of the abelian Galois group, i.e., that for c 2 E  , we have

                                     ŒA; N; N  ŒA; N; NN˚ .c/:

The isomorphism is given by N˚ .c/W A ! A (note that NE=F .N˚ .c// D NE  =Q .c/ 2 Q ,
so that N is respected).                                                           2

     For K  A  E;f
                      , sufficiently small, the objects .A; ; N/ have no nontrivial automor-
phisms. In the proposition, pass to the inductive limit over k, up to the algebraic closure of
E  , then pass to the projective limit over K. One finds the following variant.

P ROPOSITION 9.28 Let EN  be an algebraic closure of E  and let M C be the set of iso-
                                 N N/ (A as above, over k,  an E-linear isomorphism of
morphism classes of objects .A; ;
                         N
V .A/ with V ˝ Af , and  a homogeneous polarization). Then the action of Gal.EN  =E  /
b
on M C is abelian, and for e 2 A                                
                                E  with finite component ef 2 AE  , we have

                                          N N D ŒA; ;
                                                      N N˚ .ef / N:
                                               
                                 '.e/ ŒA; ;
CHAPTER II. THE ARITHMETIC THEORY                                                              88


10     The fundamental theorem of complex multiplication
The first three subsections are based on Tate 1981 and the last section on Deligne 1981.
    We begin by reviewing some notation. We let b     Z D lim Z=mZ and Af D b       Z ˝ Q. For a
number field k, Af;k D Af ˝Q k is the ring of finite adèles and Ak D Af;k  .k ˝Q R/ is
the full ring of adèles. When k is a subfield of C, k ab and k al denote respectively the largest
abelian extension of k in C and the algebraic closure of k in C. Complex conjugation is
denoted by .
    For a number field k, reck W A k
                                      ! Gal.k ab =k/ is the usual reciprocity law and artk is its
reciprocal: a prime element corresponds to the inverse of the usual (arithmetic) Frobenius.
In more detail, if a 2 A  f;k
                               has v-component a prime element av in kv and w-component
aw D 1 for w ¤ v, then

                         artk .a/.x/  x 1=N.v/    mod pv ;      x 2 Ok :
                                                                rk
When k is totally imaginary, artk factors into Ak
                                                   ! Af;k
                                                           ! Gal.k ab =k/; we usually write
                                                           Z  A is the homomorphism
artk for rk . The cyclotomic character  D cyc W Aut.C/ ! b             f
such that  D  . / for every root of 1 in C. The composite

                                      artk ı cyc D Verk=Q ;                                 (79)

the Verlagerung map Gal.Qal =Q/ab ! Gal.Qal =k/ab .

Statement of the Theorem
Let A be an abelian variety over C, and let E be a subfield of End.A/ ˝ Q of degree 2 dim A
over Q. The representation of E on the tangent space to A at zero is of the form ˚'2˚ '
with ˚ a subset of Hom.E; C/. A Riemann form for A is a Q-bilinear skew-symmetric
form on H1 .A; Q/ such that

                       .x; y/ 7!     .x; iy/W H1 .A; R/  H1 .A; R/ ! R

is symmetric and positive definite. We assume that there exists a Riemann form           compat-
ible with the action of E in the sense that, for some involution E of E,

                    .ax; y/ D      .x; .E a/y/;   a 2 E;      x; y 2 H1 .A; Q/:

Then E is a CM-field, and ˚ is a CM-type on E, i.e., Hom.E; C/ D ˚ [˚ (disjoint union).
The pair .A; E ,! End.A/ ˝ Q/ is said to be of CM-type. For simplicity, we assume that
E \ End.A/ D OE , the full ring of integers in E.
    Let C˚ be the set of complex-valued functions on ˚, and embed E into C˚ through
the natural map a 7! .'.a//'2˚ . There then exist a Z-lattice a in E stable under OE ,
an element t 2 E  , and an OE -linear analytic isomorphism W C˚ =˚.a/ ! A such that
  .x; y/ D TrE=Q .tx  E y/ where, in the last equation, we have used  to identify H1 .A; Q/
with a ˝ Q D E. The variety is said to be of type .E; ˚ I a; t / relative to . The type
determines the triple .A; E ,! End.A/ ˝ Q; / up to isomorphism. Conversely, the triple
determines the type up to a change of the following form: if  is replaced by  ı a 1 ,
                                               t
a 2 E  , then the type becomes .E; ˚ I aa; aa  / (see 3.17).
    Let  2 Aut.C/. Then E ,! End .A/ induces a map E ,! End0 .A/, so that A
                                         0

also has complex multiplication by E. The form is associated with a divisor D on A,
CHAPTER II. THE ARITHMETIC THEORY                                                             89


and we let  be the Riemann form for A associated with D. It has the following
characterization: after multiplying with a nonzero rational number, we can assume that it
takes integral values on H1 .A; Z/; define m to be the pairing Am  Am ! m , .x; y/ 7!
exp. 2 i m.x;y/ /; then . /m .x; y/ D  . m .x; y// for all m.
    In the next section we shall define for each CM-type .E; ˚ / a map f˚ W Aut.C/ !
Af;E
      =E  such that

                            f˚ . /  f˚ . / D cyc . /E  ;   all  2 Aut.C/:

We can now state the fundamental theorem of complex multiplication.

T HEOREM 10.1 (S HIMURA , TANIYAMA , W EIL L ANGLANDS , D ELIGNE , TATE , ET AL )
Suppose A has type .E; ˚ I a; t / relative to the uniformization W C˚ =a ! A. Let  2
Aut.C/, and let f 2 A
                     f;E
                          lie in f˚ . /.
  (a) The variety A has type
                                                     tcyc . / 
                                       E;  ˚ I f a;
                                                      f  f
       relative to  0 say.
  (b) It is possible to choose  0 so that

                                                                              
                     Af;E                      Af;E =a ˝ b
                                                         Z ' E=a                    Ators
                      f                                                                 

                                                                              0
                     Af;E                    Af;E =.f a ˝ b
                                                          Z/ ' E=f a                 Ators

       commutes, where Ators denotes the torsion subgroup of A (and then  0 is uniquely
       determined),

    We now restate the theorem in more invariant form. Let
                    def                 1
                 TA D lim Am .C/ ' lim. m H1 .A; Z/=H1 .A; Z// ' H1 .A; b
                                                                        Z/

(limit over all positive integers m), and let
                                       def
                                 Vf A D TA ˝Z Q ' H1 .A; Q/ ˝Q Af :

Then     gives rise to a pairing

                                   f D lim     m W Vf A  Vf A ! Af .1/;

where Af .1/ D .lim m .C// ˝ Q.

T HEOREM 10.2 Let A have type .E; ˚ /; let  2 Aut.C/, and let f 2 f˚ . /.
  (a) A is of type .E;  ˚ /;
  (b) there is an E-linear isomorphism ˛W H1 .A; Q/ ! H1 .A; Q/ such that
                    . /
          i)   . fcycf x; y/ D . /.˛x; ˛y/;            x; y 2 H1 .A; Q/;
CHAPTER II. THE ARITHMETIC THEORY                                                                     90


         ii) the32 diagram
                                                            f
                                               Vf .A/                   Vf .A/
                                                                            ˛˝1
                                                            
                                                                        Vf .A/
             commutes.
L EMMA 10.3 The statements (10.1) and (10.2) are equivalent.
                                                                r                           
P ROOF. Let  and  0 be as in (10.1), and let 1 W E ! H1 .A; Q/ and 10 W E ! H1 .A; Q/
be the E-linear isomorphisms induced by  and  0 . Let  D cyc . /=f  f — it is an
element of E  . Then

                                    .1 .x/; 1 .y// D TrE=Q .tx  y/
                              . /.10 .x/; 10 .y// D TrE=Q .tx  y/

and
                                                     1
                                       Af;E                Vf .A/
                                           f                        
                                                     10
                                       Af;E                Vf .A/

commutes. Let ˛ D 10 ı 1 1 ; then

                   . /.˛x; ˛y/ D TrE=Q .t1 1 .x/  1 1 .y// D                .x; y/

and (on Vf .A/),
                             D 10 ı f ı 1 1 D 10 ı 1 1 ı f D ˛ ı f:
Conversely, let ˛ be as in (10.2) and choose 10 so that ˛ D 10 ı 1 1 . The argument can be
reversed to deduce (10.1).                                                                  2


Definition of f˚ . /
Let .E; ˚/ be a CM-pair with E a field. In (9.9) we saw that N˚ gives a well-defined homo-
morphism Aut.C=E  / ! A  f;E
                               =E  . In this subsection, we extend this to a homomorphism
on the whole of Aut.C/.
    Choose an embedding E ,! C; and extend it to an embedding i W E ab ,! C. Choose
elements w 2 Aut.C/, one for each  2 Hom.E; C/, such that

                                    w ı i jE D ;         w D w :

For example, choose w for  2 ˚ (or any other CM-type) to satisfy the first equation,
and then define w for the remaining  by the second equation. For any  2 Aut.C/,
w1  w ı i jE D w1 ı jE D i . Thus i 1 ı w1  w ı i 2 Gal.E ab =E/, and we can
define F˚ W Aut.C/ ! Gal.E ab =E/ by
                                       Y
                             F˚ . / D    i 1 ı w '1  w' ı i:
                                               '2˚
   32 Note that both f 2 A   and the E-linear isomorphism ˛ are uniquely determined up to multiplication
                          f;E
                    
by an element of E . Changing the choice of one changes that of the other by the same factor.
CHAPTER II. THE ARITHMETIC THEORY                                                         91


L EMMA 10.4 The element F˚ is independent of the choice of fw g.

P ROOF. Any other choice is of the form w0 D w h , h 2 Aut.C= iE/. Thus F˚ . / is
changed by i 1 ı . '2˚ h '1 h' / ı i . The conditions on w and w 0 imply that h D h , and
                    Q
it follows
    Q            Q inside product is 1 because  permutes the unordered pairs f'; 'g and
           that the
so '2˚ h' D '2˚ h ' .                                                                      2

L EMMA 10.5 The element F˚ is independent of the choice of i (and E ,! C).

P ROOF. Any other choice is of the form i 0 D  ı i ,  2 Aut.C/. Take w0 D w ı  1 , and
then                         Y
                  F˚0 . / D    i 0 1 ı . w '1  w'  1 / ı i 0 D F˚ . /:              2

   Thus we can suppose E  C and ignore i ; then
                              Y
                    F˚ . / D    w '1  w' mod Aut.C=E ab /;
                                  '2˚

where the w are elements of Aut.C/ such that

                                  w jE D ;      w D w :

P ROPOSITION 10.6 For any  2 Aut.C/, there is a unique f˚ . / 2 A
                                                                   f;E
                                                                       =E  such that
  (a) artE .f˚ .// D F˚ . /;
  (b) f˚ ./  f˚ ./ D . /E  ,  D cyc .

P ROOF. Since artE is surjective, there is an f 2 A
                                                   f;E
                                                       =E  such that artE .f / D F˚ . /.
We have

                           artE .f  f / D artE .f /  artE .f /
                                         D artE .f /  artE .f / 1
                                         D F˚ . /  F˚ . /
                                         D VerE=Q . /;

where VerE=Q W Gal.Qal =Q/ab ! Gal.Qal =E/ab is the transfer (Verlagerung) map. As VerE=Q D
artE ı , it follows that f  f D . /E  modulo Ker.artE /. Lemma 9.6 shows that 1 C 
acts bijectively on Ker.artE /, and so there is a unique a 2 Ker.artE / such that a  a D
.f  f =.//E  ; we must take f˚ . / D f =a.                                         2

R EMARK 10.7 The above definition of f˚ . / is due to Tate. The original definition, due
to Langlands, was more direct but used the Weil group (see my notes Abelian Varieties with
Complex Multiplication (for Pedestrians), 7.2).

P ROPOSITION 10.8 The maps f˚ W Aut.C/ ! A
                                          f;E
                                              =E  have the following properties:
  (a) f˚ ./ D f ˚ . /  f˚ . /;
  (b) f˚. 1 jE / ./ D f˚ . / if E D E;
  (c) f˚ ./ D 1.
CHAPTER II. THE ARITHMETIC THEORY                                                                          92


P ROOF. Let f D f ˚ . /  f˚ . /. Then
                                                 Y
           artE .f / D F ˚ . /  F˚ . / D           w 1'   w '  w '1   w' D F˚ .  /
                                                 '2˚

and f  f D ./. /E  D .  /E  . Thus f satisfies the conditions that determine
f˚ . /. This proves (a), and (b) and (c) can be proved similarly.                   2

   Let E  be the reflex field for .E; ˚ /, so that Aut.C=E  / D f 2 Aut.C/ j  ˚ D ˚ g.
                  def S
Then ˚ Aut.C=E/ D '2˚ '  Aut.C=E/ is stable under the left action of Aut.C=E  /, and
we write                            [
               Aut.C=E/˚ 1 D              Aut.C=E  /    .disjoint union/:
The set Q    D f jE  g is a CM-type for E  , and .E  ; / is the reflex of .E; ˚ /. The
map a 7!        2     .a/W E  ! C factors through E and defines a morphism of algebraic
              
tori N˚ W T E ! T E . The fundamental theorem of complex multiplication over the re-
flex field states the following: let  2 Aut.C=E  /, and let a 2 A
                                                                   f;E 
                                                                         =E  be such that
artE  .a/ D ; then (10.1) is true after f has been replaced by N˚ .a/ (see Theorem 9.10;
also Shimura 1971, Theorem 5.15; the sign differences result from different conventions
for the reciprocity law and the actions of Galois groups). The next result shows that this is
in agreement with (10.1).
P ROPOSITION 10.9 For any  2 Aut.C=E  / and a 2 A
                                                   f;E 
                                                         =E  such that artE  .a/ D
 jE ab , N˚ .a/ 2 f˚ . /.
                  ˚ into orbits, ˚ D [j ˚j , for the left action of Aut.C=E  /. Then
P ROOF. Partition S
Aut.C=E/˚ 1 D j Aut.C=E/˚j 1 , and

  Aut.C=E/˚j 1 D Aut.C=E/.j 1 Aut.C=E  // D .HomE .Lj ; C/ ı j 1 / Aut.C=E  /;

where j is any element of Aut.C/ such that j jE 2 ˚j and Lj D .j 1 E  /E. Thus
N˚ .a/ D bj , with bj D NmLj =E .j 1 .a//. Let
         Q

                              Y
                    Fj . / D    w '1  w' .mod Aut.C=E ab //:
                                     '2˚j

We begin by showing that Fj . / D artE .bj /. The basic properties of Artin’s reciprocity
law show that

                   injective                              j 1                         NmLj =K
    A
     f;E
                                     A
                                      f;L
                                                                         A
                                                                          f;L
                                                                                                        A
                                                                                                         f;K
                                             j                                 j

        artE                             artLj                              artLj                         artK
                Vj Lj =E                                  adj 1                      restriction
                                                 1
 Gal.E ab =E/                  j Gal.Lab
                                       j =Lj /j                     Gal.Lab
                                                                          j =Lj /                    Gal.K ab =K/

    commutes. Therefore artE .bj / is the image of artE  .a/ by the three maps in the bottom
row of the diagram. Consider ft' j t' D w' j 1 ; ' 2 ˚j g; this is a set of coset repre-
sentatives for j Aut.C=Lj /j 1 in Aut.C=E  /, and so Fj . / D '2˚j j 1 t '1  t' j D
                                                                    Q

j 1 V ./j mod Aut.C=E      ab
                           Q /.
    Thus artE .N˚ .a// D artE .bj / D Fj . / D F˚ . /. As N˚ .a/N˚ .a/ 2 cyc . /E 
                                         Q
(see 9.7), this shows that N˚ .a/ 2 f˚ . /.                                                2
CHAPTER II. THE ARITHMETIC THEORY                                                           93


Proof of Theorem 10.2 up to an element of order 2
The variety A has type .E;  ˚ / because  ˚ describes the action of E on the tangent
space to A at zero. Choose any E-linear isomorphism ˛W H1 .A; Q/ ! H1 .A; Q/. Then

                                                       .˛˝1/   1
                                      
                             Vf .A/ ! Vf .A/            !         Vf .A/

is an Af;E -linear isomorphism, and hence is multiplication by some g 2 A
                                                                         f;E
                                                                             ; thus

                                          .˛ ˝ 1/ ı g D :

L EMMA 10.10 For this g, we have
                         . /      
                    .˛ /         x; y D . /.x; y/;            all x; y 2 Vf .A/:
                          g  g
P ROOF. By definition,

                       . /.x; y/ D  . .x; y//                  x; y 2 Vf .A/
                       .˛ /.˛x; ˛y/ D         .x; y/       x; y 2 Vf .A/:

On replacing x and y by gx and gy in the second inequality, we find that

                       .˛ /.x; y/ D         .gx; gy/ D           ..g  g/x; y/:

As  . .x; y// D . / .x; y/ D      .. /x; y/, the lemma is now obvious.                  2

R EMARK 10.11        (a) On replacing x and y with ˛x and ˛y in (10.10), we obtain the
     formula                         . /       
                                              x; y D . /.˛x; ˛y/:
                                     g  g
  (b) On taking x; y 2 H1 .A; Q/ in (10.10), we can deduce that cyc . /=g  g 2 E  ; there-
      fore g  g  cyc . / modulo E  .

     The only choice involved in the definition of g is that of ˛, and ˛ is determined up to
multiplication by an element of E  . Thus the class of g in A
                                                              f;E
                                                                  =E  depends only on A and
. In fact, it depends only on .E; ˚ / and , because any other abelian variety of type .E; ˚ /
is isogenous to A and leads to the same class gE  . We define g˚ . / D gE  2 A   f;E
                                                                                         =E  .

P ROPOSITION 10.12 The maps g˚ W Aut.C/ ! A
                                           f;E
                                               =E  have the following properties:
  (a) g˚ ./ D g ˚ . /  g˚ . /;
  (b) g˚. 1 jE / ./ D g˚ . / if E D E;
  (c) g˚ ./ D 1;
  (d) g˚ ./  g˚ ./ D cyc . /E  .

P ROOF. (a) Choose E-linear isomorphisms ˛W H1 .A; Q/ ! H1 .A; Q/ and ˇW H1 .A; Q/ !
H1 . A; Q/, and let g D .˛ ˝ 1/ 1 ı  and g D .ˇ ˝ 1/ 1 ı  so that g and g represent
g˚ . / and g ˚ ./ respectively. Then

                     .ˇ˛/ ˝ 1 ı .g g/ D .ˇ ˝ 1/ ı g ı .˛ ˝ 1/ ı g D  ;
CHAPTER II. THE ARITHMETIC THEORY                                                                94


which shows that g g represents g˚ .  /.
                                                                          1
    (b) If .A; E ,! End.A/ ˝ Q/ has type .E; ˚ /, then .A; E ! E ! End.A/ ˝ Q/ has
type .E; ˚ 1 /. The formula in (b) can be proved by transport of structure.
    (c) Complex conjugation W A ! A is a homeomorphism (relative to the complex topol-
ogy) and so induces an E-linear isomorphism 1 W H1 .A; Q/ ! H1 .A; Q/. The map 1 ˝
1W Vf .A/ ! Vf .A/ is  again, and so on taking ˛ D 1 , we find that g D 1.
    (d) This was proved in (10.11d).                                                  2

   Theorem (10.2) (hence also 10.1) becomes true if f˚ is replaced by g˚ . Our task is to
show that f˚ D g˚ . To this end we set

                               e˚ . / D g˚ . /=f˚ . / 2 A     
                                                            f;E =E :                            (80)

P ROPOSITION 10.13 The maps e˚ W Aut.C/ ! A
                                           f;E
                                               =E  have the following properties:
  (a) e˚ ./ D e ˚ . /  e˚ . /;
     (b) e˚. 1 jE / ./ D  e˚ . / if E D E;
     (c) e˚ ./ D 1;
     (d) e˚ ./  E e˚ . / D 1;
     (e) e˚ ./ D 1 if  ˚ D ˚.

P ROOF. Statements (a), (b), and (c) follow from (a), (b), and (c) of (10.8) and (10.12), and
(d) follows from (10.6b) and (10.12d). The condition  ˚ D ˚ in (e) means that  fixes
the reflex field of .E; ˚ / and, as we observed in the preceding subsection, the fundamental
theorem is known to hold in that case, which means that f˚ . / D g˚ . /.                  2

P ROPOSITION 10.14 Let F be the largest totally real subfield of E; then e˚ . / 2 A
                                                                                    f;F
                                                                                        =F 
and e˚ ./2 D 1; moreover, e˚ . / depends only on the effect of  on E  , and is 1 if
 jE  D id.

P ROOF. Recall (1.16) that  fixes E  if and only if  ˚ D ˚, in which case (10.13e) shows
that e˚ ./ D 1. Replacing  by  1  in (a), we find that e˚ . / D e˚ . / if  ˚ D  ˚,
i.e., e˚ ./ depends only on the restriction of  to the reflex field of .E; ˚ /. From (b) with
 D , we find using ˚ D ˚ E that e˚ . / D e˚ . /. Putting  D  in (a) and using (c)
we find that e˚ ./ D e˚ . /; putting  D  in (a) and using (c) we find that e˚ . / D
e˚ . /. Since  and   have the same effect on E  , we conclude e˚ . / D e˚ . /. Thus
e˚ . / 2 .Af;E
                 =E  /hE i , which equals A
                                             f;F
                                                 =F  by Hilbert’s Theorem 90.33 Finally, (d)
shows that e˚ ./2 D 1.                                                                       2

C OROLLARY 10.15 Part (a) of (10.1) is true; part (b) of (10.1) becomes true when f is
replaced by ef with e 2 A
                         f;F
                             , e 2 D 1.
     33 The cohomology sequence of the sequence of Gal.E=F /-modules


                                    1 ! E  ! A           
                                               f;E ! Af;E =E ! 1

is
                                           Gal.E=F /                           .FT 5.22)
                 1 ! F  ! A       
                            f;F ! .Af;E =E /          ! H 1 .Gal.E=F /; E  /      D        0
CHAPTER II. THE ARITHMETIC THEORY                                                            95


P ROOF. Let e 2 e˚ . /. Then e 2 2 F  and, since an element of F  that is a square locally
at all finite primes is a square (CFT, VIII, 1.1), we can correct e to achieve e 2 D 1. Now
(10.1) is true with f replaced by ef , but e (being a unit) does not affect part (a) of (10.1).2

    It remains to show that:

                    for all CM-fields E and CM-types ˚ on E, e˚ D 1:                       (81)

Completion of the proof (following Deligne)
As above, let .E; ˚/ be a CM pair, and let e˚ . / D g˚ . /=f˚ . / be the associated element
of A
    f;E
        =E  . Then, as in (10.14, 10.15),

                           e˚ . / 2 2 .Af;F /=2 .F /;    2 Aut.C/:

    Let
                 e 2 2 .Af;F /, e D .ev /v , ev D ˙1, v a finite prime of F
be a representative for e˚ . /. We have to show that the ev ’s are all 1 or all C1. For
this, it suffices, to show that for, for any prime numbers `1 and `2 , the image of e˚ . / in
2 .F`1  F`2 /=2 .F / is trivial. Here F` D F ˝Q Q` .
     In addition to the properties (a–e) of (10.13), we need:
   (f) let E 0 be a CM-field containing E, and let ˚ 0 be the extension of ˚ to E 0 ; then for
        any  2 Aut.C/,
                                 e˚ ./ D e˚ 0 ./ (in A         0
                                                          f;E 0 =E ).                     (82)

To prove this, one notes that the same formula holds for each of f˚ and g˚ : if A is of type
                 def
.E; ˚ / then A0 D A˝E E 0 is of type .E 0 ; ˚ 0 /. Here A0 D AM with M D HomE -linear .E 0 ; E/
(cf. 7.33).
     Note that (f) shows that e˚ 0 D 1 H) e˚ D 1, and so it suffices (81) for E Galois over
Q (and contained in C).
     We also need:
  (g) denote by Œ˚ the characteristic function of ˚  Hom.E; C/; then
                     X                      Y
                         ni Œ˚i  D 0 H)          e˚i . /ni D 1 for all  2 Aut.C/:
                       i                       i

This is a consequence of Deligne’s theorem that all Hodge classes on abelian varieties are
absolutely Hodge, which tells us that the results on abelian varieties with complex multi-
plication proved above extend to CM-motives. The CM-motives are classified by infinity
types rather than CM-types, and (g) just says that the e attached to the trivial CM-motive is
1. This will be explained in the next chapter.
    We make (d) (of 10.13) and (g) more explicit. Recall that an infinity type on E is a
function W Hom.E; C/ ! Z that can be written as a finite sum of CM-types (see 4). Now
(g) allows us to define e by linearity for  an infinity type on E. Moreover,

                                         e2 D e2 D 0;

so that e depends only on the reduction modulo 2 of , which can be regarded as a function

                                    W Hom.E; C/ ! Z=2Z;
                                    N
CHAPTER II. THE ARITHMETIC THEORY                                                                 96


such that either (weight 0)
                                      .'/ C N
                                      N     .'/ D 0 for all '                                  (83)
or (weight 1)
                                      .'/ C N
                                      N     .'/ D 1 for all '.
   We now prove that eN D 1 if N is of weight 0. The condition (83) means that N
                                                                                 .'/ D
.'/, and so N arises from a function qW Hom.F; C/ ! Z=2Z:
N

                                                .'/ D q.'jF /:
                                                N

We write eq D eN . When E is a subfield of C Galois over Q, (b) implies that there exists an
e. / 2 2 .Af;F /=2 .F / such that34
                                    Y
                        eq . / D                   ' 1 .e. //q.'/ ,  2 Aut.C/:
                                          'WF !C

Write e./ D e F ./ to denote the dependence of e on F . It follows from (f), that for any
totally real field F 0 containing F ,
                                                                   0
                                      e F . / D NmF 0 =F e F . /:

There exists a totally real field F 0 , quadratic over F , and such that all primes of F dividing
`1 or `2 remain prime in F 0 . The norm maps 2 .F2;` / ! 2 .F1 ;` / are zero for ` D `1 ; `2 ,
and so e F ./ projects to zero in 2 .F`1 /  2 .F`2 /=2 .F /. Therefore eq . / projects to
zero in 2 .F`1  F`2 /=2 .F /. This being true for every pair .`1 ; `2 /, we have eq D 1.
    We now complete the proof of (81). We know that eN depends only on the weight of
N
, and so,
         p for ˚ a CM-type, e˚ . / depends only on  . Inp      calculating e˚ . /, we may take
E D Q.       1/P and ˚ to be one of the  ptwo  CM-types   on QŒ     1. We know (see 10.14) that
                               
e˚ . / depends only on jE D QŒ            1. But e˚ .1/ D 1 D e˚ ./ by (10.13c).

A SIDE 10.16 Throughout, should allow E to be a CM-algebra. Should restate Theorem
10.2 with C replaced by Qal ; then replace C with Qal throughout the proof (so  is an
automorphism of Qal rather than C).




   34 For each 'W F ! C, choose an extension (also denoted ') of ' to E. Then


                                                .' 0 /' 0 D
                                  X                            X
                           N D                 N                       q.'/.' C '/
                                    ' 0 WE !C                  'WF !C

and so
                           def
                                                     ./q.'/ D              ' 1 .e1C .//q.'/
                                          Y                        Y
                     eq ./ D eN ./ D      e
                                            ' .1C/'                    '
— we can take e. / D e1C ./.
Chapter III

CM-Motives

 ˘ Explain and prove Deligne’s theorem that Hodge classes on abelian varieties are ab-
   solutely Hodge (at least in the CM-case).
 ˘ Construct the category of abelian motives (with complex multiplication) over any
   field of characteristic zero. Observe that over C, the category coincides that defined
   in Chapter I.
 ˘ Construct the Taniyama group (Langlands/Tate) and observe that the fundamental
   theorem shows that it is the Tannaka group for the category of CM-motives over Q
   (with additional structure).
 ˘ Re-interprete the earlier results more motivically.




                                          97
Chapter IV

Applications

˘ Abelian varieties over finite fields.
˘ Zeta function of abelian varieties of CM-type (even over Q, using the Taniyama
  group).
˘ Hilbert’s 12th problem.
˘ Periods, including the period torsor.
˘ Algebraic Hecke characters (“. . . the connection between [algebraic Hecke charac-
  ters] and abelian varieties with complex multiplication appears to be so close that it
  can hardly be accidental; and any future arithmetical interpretation of the [algebraic
  Hecke characters], corresponding to the interpretation given by class field theory for
  the characters of finite order of the idèle class group, ought to take complex multipli-
  cation into account” Weil (Weil (1956a)1956a, p. 18).)
˘ Summary of applications to modular functions and forms.
˘ And so on (AVCM (for pedestrians), Blasius 1986, Colmez 1993, Schappacher 1988,
  Shimura 1998, Yoshida 2003, . . . ).




                                           98
Appendix A

Additional Notes; Solutions to the
Exercises

E XERCISE 1.11
Let E1 be a Galois closure of E  over Q with Galois group G, and extend ˚ to a CM-type
˚1 on E1 . Note that E1 D F1 Œ˛ and that ˚1 D f' j =.'.˛// > 0g. Let

                                 H D f 2 G j ˚1  D ˚1 g:

Then H is the set of  2 G such that  ˛=˛ is totally positive, and so it is the subgroup of G
fixing E  if and only the conditions (a) and (b) in the exercise hold. Now apply Corollary
1.10.

E XERCISE 2.9
Because E  is a product of separable field extensions, the trace pairing

                            .x; y/ 7! TrE  =Q .xy/W E   E  ! Q

is nondegenerate. For any Q-bilinear form W E   E  ! Q, the map x 7!             .x; 1/W E  !
Q is Q-linear, and so
                                  .x; 1/ D TrE  =Q .˛x/
for a unique ˛ 2 E  , which lies in E  if is nondegenerate. If           satisfies (a) (of 2.9),
then
                            .x; y/ D . N yx; 1/ D TrE  =Q .˛x N
                                                              y/:
Conversely, for any ˛ 2 E  , .x; y/ 7! TrE  =Q .˛x N
                                                      y/ is a nondegenerate Q-bilinear form
satisfying (a) .
    Now let
                                   .x; y/ D TrE  =Q .˛x N
                                                         y/                             (84)
for some ˛ 2 E  . We have

        .y; x/ D TrE  =Q .˛y Nx/ D TrE  =Q .˛N N
                                                yx/ (as TrE  =Q . Nx/ D TrE  =Q .x/), and so
        .y; x/ D TrE  =Q .. N˛/x N
                                 y/:

On comparing this with (84), we see that (b) holds for       if and only if ˛ D N˛.

                                               99
APPENDIX A. ADDITIONAL NOTES; SOLUTIONS TO THE EXERCISES                                 100


    Under the isomorphism
                                   r
                      E  ˝Q R ! a ˝ r 7! .: : : ; r  '.a/; : : :/C˚ ;
                                                                p
multiplication by J˚ corresponds
                          P      to multiplication by i D           1 (this is how we defined
J˚ ), and corresponds to '2˚ ' , where

                                                               def
                           ' .x; y/ D TrC=R .˛' x N
                                                 y/;       ˛' D '.˛/:

Now
                         ' .ix; iy/ D TrC=R .˛'  ix  iy/ D         ' .x; y/;

and so (c) holds automatically. Finally, because ˛' is totally imaginary,

                                   ' .x; y/ D ˛' .x yN   Nxy/;

and so
                                  ' .x; ix/ D   .i ˛' /.2x Nx/:
This is > 0 for all nonzero x if and only if =.˛' / > 0.

E XERCISE 3.10
Appendix B

Summary

                                                            def
 1 Let A be an abelian variety over a field k. Then End0 .A/ D End.A/˝Z Q is a semisimple
Q-algebra of reduced degree  2 dim A. When equality holds, we say that A has complex
multiplication over k (or be a CM abelian variety over k, or an abelian variety of CM-type
over k).
 2 When A has complex multiplication over k, all maximal étale Q-subalgebras of End0 .A/
have degree 2 dim A over Q. We say that A has complex multiplication by E over k when E
is an étale subalgebra of End0 .A/ of degree 2 dim A. When, in addition, End.A/ \E D OE ,
the ring of integers in E, we say that A has complex multiplication by OE over k.
 3 Let A have complex multiplication by E over a field k of characteristic zero, and assume
that k contains all conjugates of E. Then
                                    M
                         Tgt0 .A/ '     k' (as E ˝Q k-modules);
                                    '2˚

where k' is a one-dimensional k-vector space on which E acts through ', and ˚ is a subset
of HomQ-alg .E; k/ such that

              HomQ-alg .E; k/ D ˚ t ˚ for all complex conjugations  on k.             (85)

Conversely, when k is algebraically closed, every pair .E; ˚ / satisfying (85) arises from an
abelian variety (unique up to an E-isogeny).
 0.17 Let k  K be algebraically closed fields of characteristic zero. The functor A
AK from abelian varieties over k to abelian varieties over K is fully faithful, and it is an
equivalence on the subcategories of CM abelian varieties.
 4 A number field is a CM-field if it admits a unique nontrivial complex conjugation,
and a CM-algebra is a finite product of CM-fields. A CM-type on E is a subset ˚ of
HomQ-alg .E; N
            Q/ satisfying

                               HomQ-alg .E; k/ D ˚ t ˚ E :                             (86)

Here NQ is an algebraic closure of Q. A CM-pair .E; ˚ / is a CM-algebra together with a
CM-type. Let L be an étale Q-algebra and ˚ subset of HomQ-alg .L; Q/;  N then ˚ satisfies
(85) if and only if there exists a CM-pair .E; ˚0 / with E  L such that

                                   ˚ D f' j 'jE 2 ˚0 g:

                                            101
APPENDIX B. SUMMARY                                                                                 102


 5 The reflex field of a CM-pair .E; ˚ / is the smallest subfield E  of N
                                                                         Q for which there
                  
exists an E ˝Q E -module V such that
                                  M
                       V ˝E  N
                              Q'       QN ' (as E ˝Q N Q-modules);
                                        '2˚


where Q N ' is a one-dimensional Q-vector
                                 N        space on which E acts through '. The E ˝Q E  -
                                                                         
module V is unique up to a nonunique isomorphism. Let T E and T E be the algebraic
tori over Q with Q-points E  and E  respectively. The reflex norm is the homomorphism
         
N˚ W T E ! T E such that, for any Q-algebra R and a 2 .E  ˝Q R/ ,

                                 N˚ .a/ D detE ˝Q R .ajV ˝Q R/

(determinant of x 7! axW V ˝Q R ! V ˝Q R regarded as an E ˝Q R-module). For any
number field k  N
                 Q containing all conjugates of E and element (or ideal, . . . ) a of k,
                                           Y
                       N˚ .Nmk=E  a/ D         ' 1 .Nmk='E a/:
                                                   '2˚

 6 (S HIMURA -TANIYAMA FORMULA ) Let A be an abelian variety with complex multipli-
cation by OE over a number field k containing the conjugates of E. Let p be a prime ideal
of Ok at which A has good reduction. Then
  (a) there exists an element  2 OE inducing the Frobenius endomorphism1 on the re-
       duction A0 of A, and
  (b) the ideal generated by  factors as
                                                  Y
                                     OE D              ' 1 .Nmk='E p/;
                                                  '2˚

       where ˚  Hom.E; k/ is the CM-type of A.

 7 Let A have complex multiplication by OE over k. For any ideal a in OE , there is a
“smallest quotient” ˛ a W A ! Aa , unique up to unique isomorphism, such that aW A ! A
factors through ˛ a for all a 2 a;2 it is any isogeny, and there is an OE -structure on Aa for
which ˛ a is an OE -isogeny. Any such OE -isogeny is called an a-multiplication.

 8 Let A be an abelian variety with complex multiplication by OE over a sufficiently large
number field k Galois over E  . Let P be a prime ideal of Ok at which A has good re-
                       def
duction. Assume .p/ D P \ Z is unramified in E and that P is unramified over E  . Let
p D P \ E  , and let  D .P; k=E  / (the Frobenius automorphism at P); then
                                                                                            .q/
  (a) there exists an a-multiplication ˛W A ! A whose reduction ˛0 W A0 ! A0                     is the
      q-power Frobenius map, q D .OE  W p/;
  (b) moreover, a D N˚ .p/.
    1 Let q D .O W p/ be the order of the residue field. By the Frobenius endomorphism of A we mean the
                 k                                                                         0
q-power Frobenius map 0 W A0 ! A0 .
    2 In scheme-theoretic terms, it is the quotient of A by

                                              def \
                                    Ker.˛ a / D       Ker.aW A ! A/:
APPENDIX B. SUMMARY                                                                      103


 9 (F UNDAMENTAL THEOREM OVER THE REFLEX FIELD : IDEAL VERSION ) Let A be an
abelian variety with complex multiplication by OE over Q,N and fix an integer m > 0. Then,
                                
there exists a modulus m for E such that the following hold:
  (a) for each fractional ideal a of E  prime to m, there exists an ideal a of OE and an
       a-multiplication ˛W A ! A, where  D .a ; Em =E  /, such that

                                  ˛.x/ D x;     for all x 2 Am I

  (b) the ideal a is determine by a up to a principal ideal in i.Em;1 /, and

                                   a  N˚ .a / mod i.Em;1 /:

 10 (F UNDAMENTAL THEOREM OVER THE REFLEX FIELD : ID ÈLE VERSION ) Let A be an
abelian variety with complex multiplication by E over N  Q, and let  2 Gal. N
                                                                            Q=E  /. For any
s 2 Af;E
          with art.s/ D jE ab , there is a unique E-isogeny ˛W A ! A such that

                            ˛.N˚ .s/  x/ D x for all x 2 Vf A:

If s is replaced with as, a 2 E  , then ˛ must be replaced by ˛a 1 .

 11 (F UNDAMENTAL THEOREM OVER THE REFLEX FIELD : UNIFORMIZATION VERSION )
Let .A; i W E ,! End0 .A// be an abelian variety with complex multiplication over C, and let
 be a polarization of .A; i /. Recall (3.11, 3.17) that the choice of a basis element e0 for
H0 .A; Q/ determines a uniformization W C˚ ! A.C/, and hence a quadruple .E; ˚ I a; t /,
called the type of .A; i; / relative to . Let  be an automorphism of C fixing E  . For
                                                                                            0
any s 2 A  f;E 
                  such that artE  .s/ D jE ab , there is a unique uniformization  0 W C˚ !
.A/.C/ of A such that
  (a) .A; i; / has type .E; ˚ I f a; t  cyc . /=f fN/ where f D N˚ .s/ 2 A f;E
                                                                                    I
  (b) the diagram
                                                0
                                       E=a            A.C/
                                          f               
                                                00
                                       E=f a          A.C/

      commutes, where 0 .x/ D ..'x/'2˚ / and 00 .x/ D  0 ..'x/'2˚ 0 /:

 12 (F UNDAMENTAL THEOREM OVER THE REFLEX FIELD : MODULI VERSION ) See The-
orem 9.19

13 (F UNDAMENTAL THEOREM OVER Q: ID ÈLE VERSION ) See Theorem 10.2.

14 (F UNDAMENTAL THEOREM OVER Q: UNIFORMIZATION VERSION ) See Theorem 10.1.
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Index of definitions

Néron model, 51                             involution, 19
                                                  positive, 19
a-multiplication, 58                         isogeny, 23
a-transform, 58                              “isogeny”, 23
abelian scheme, 49
abelian variety, 26, 46                      lattice, 21
     CM, 27                                  lattice ideal, 58
     simple, 27
     with complex multiplication, 52         multiplicative type, 32
acts on through, 32                          Mumford-Tate group, 36

CM abelian variety over k, 52                nilpotent ideal, 8
CM-algebra, 11                               number field, 6
CM-field, 10                                 numerical norm, 66
CM-pair, 11
     primitive, 11                           order, 52, 55
CM-pairs
                                             potential good reduction, 51
     isomorphism of, 18
CM-type, 11, 13, 19, 27, 29, 45, 54          reduced degree, 9
CM-type .E; ˚ /, 88                          reduced module, 8
CM-type over k, 52                           reflex CM-pair, 14
complex conjugation, 6                       reflex CM-type, 14
complex multiplication, 27                   reflex field, 13, 29
     p-divisible group, 72                   reflex norm, 16
complex multiplication by E, 52              Riemann form, 35, 88
complex multiplication over k, 52                 integral, 25
complex torus, 21                                 rational, 25
contains all conjugates, 6                   Riemann pair, 22
                                                  homomorphism of, 22
degree, 55
                                                  rational, 35
Deligne torus, 34
                                             Rosati involution, 25
diagonalizable group, 32
                                             semisimple algebra, 8
étale algebra, 6
                                             Serre condition, 41
field, 6                                     Serre group, 40, 42
fix a subfield, 6                            simple algebra, 8
                                             specialization, 48
good reduction, 49                           specialization map, 50
                                             split, 33
hermitian form, 24
“homomorphism”, 23                           torus, 33
                                             totally imaginary, 10
infinity type, 38                            totally real, 10


                                       107
INDEX OF DEFINITIONS          108


type .E; ˚ I a; t /, 31, 88

unramified, 51

weight, 38
