---
id: C-0016
type: claim
title: "Rosati polarization gives an explicit all-order positive Note-367 kernel for curves over finite fields"
status: proved-unconditional
family: adjacent-unconditional
interestingness: 3
statement: >
  For every smooth projective geometrically connected curve
  \(C/\mathbf F_q\), the centered completed numerator \(X_C\) has a
  logarithmic derivative \(L_C=X_C'/X_C\) whose descended kernel
  \(Q_C(\lambda,\mu)=4[\mu L_C(\lambda)-\lambda L_C(\mu)]/
  (\mu^2-\lambda^2)\), extended confluently on the diagonal, is positive
  semidefinite at every matrix order for all \(\lambda,\mu>0\). For the
  \(q\)-power Frobenius endomorphism \(\pi\) of the Jacobian, the positive
  Rosati involution satisfies \(\pi^\dagger\pi=[q]\), forces
  \(|\alpha|^2=q\) for every characteristic root, and yields the explicit
  Gram sum over the resulting real squared frequencies.
verifier: canon/witnesses/C-0016/verify.py
depends: []
supersedes: []
sources:
  - scratch/proof-system--c0016-rosati-purity-loewner-audit/AUDIT.md
  - scratch/proof-system--c0016-rosati-purity-loewner-audit/RESULT.md
  - archive/notes/412-function-field-polarization-transfer-obstruction.md
tags: [function-fields, rosati, polarization, loewner, frobenius]
caveats: >
  The Euler-series formula for the closed-point source is used only in
  Re(z)>1/2; the spectral Gram theorem itself holds for every positive
  node. The result is confined to finite-field curves and does not
  construct or suggest a number-field polarization.
---

## Statement

With

\[
X_C(z)=e^{g(\log q)z}P_C(q^{-1/2}e^{-(\log q)z}),\qquad
L_C(z)=\frac{X_C'(z)}{X_C(z)},\qquad
H_C(x)=\frac{X_C'(\sqrt x)}{\sqrt x\,X_C(\sqrt x)},
\]

put

\[
M_C(x,y)
:=-\frac{H_C(x)-H_C(y)}{x-y},
\qquad
M_C(x,x):=-H_C'(x).
\]

Then, for all positive \(x,y\),

\[
M_C(x,y)
=\frac{m_0}{xy}
+2\sum_{\gamma>0}
\frac{m_\gamma}{(x+\gamma^2)(y+\gamma^2)}
\succeq0.
\]

For \(x_i=\lambda_i^2\),

\[
Q_C(\lambda_i,\lambda_j)
:=4\frac{\lambda_jL_C(\lambda_i)-\lambda_iL_C(\lambda_j)}
{\lambda_j^2-\lambda_i^2}
=4\lambda_i\lambda_jM_C(x_i,x_j).
\]

Thus \(Q=DMD\), \(D=\operatorname{diag}(2\lambda_i)\), at every
matrix order. In particular,

\[
Q_C(\lambda,\lambda)
=-4\lambda^2H_C'(\lambda^2).
\]

## Mechanism

Let \(J=\operatorname{Jac}(C)\), and let \(\pi\) be its \(q\)-power
Frobenius endomorphism. Milne's Rosati lemma gives, for the involution
from any polarization,

\[
\pi^\dagger\pi=[q]_J.
\]

Rosati positivity directly proves that \(\mathbf Q[\pi]\) has no
nilpotents, is a product of fields, and is stable under \(\dagger\).
After tensoring with \(\mathbf R\), \(\dagger\) is the identity on real
factors and complex conjugation on complex factors. Therefore every
complex embedding satisfies

\[
\sigma(\pi^\dagger)=\overline{\sigma(\pi)},\qquad
|\sigma(\pi)|^2=q.
\]

Write

\[
\chi_\pi(T)=\det(T-\pi\mid V_\ell J),\qquad
P_C(u)=\det(1-u\pi\mid V_\ell J)
=u^{2g}\chi_\pi(u^{-1}).
\]

Apply Milne's Theorem 11.1 to \(C/\mathbf F_{q^n}\) for every \(n\).
The Frobenius roots become \(\alpha_j^n\), so

\[
N_n=1+q^n-\sum_j\alpha_j^n.
\]

Therefore

\[
\log Z_C(u)
=-\log(1-u)-\log(1-qu)+\sum_j\log(1-\alpha_j u),
\]

which identifies the numerator with the displayed
\(\det(1-u\pi\mid V_\ell J)\). This derivation is used instead of the
article's Corollary 11.4, whose printed \(P(t)\) conflicts with its
earlier monic-\(P(X)\) convention.

The coefficients of \(P_C\) are integral, so purity identifies the
complex-conjugate root of \(\alpha\) with \(q/\alpha\), including
multiplicity; moreover \(\det\pi=q^g\). Hence

\[
P_C(u)=q^gu^{2g}P_C(1/(qu)),
\]

which makes \(X_C\) even. It is also \(2\pi i/\log q\)-periodic. Hence its
local expansions at \(0\) and \(\pi i/\log q\) are even, so the normalized
fixed roots \(+1\) and \(-1\) occur with even multiplicity and may be grouped
into reciprocal pairs.

A normalized nonexceptional pair
\(e^{\pm i\theta}\) contributes

\[
2(\cosh((\log q)z)-\cos\theta).
\]

At \(\theta=0\), every frequency \(2\pi k/\log q\), including zero, is
doubled per reciprocal pair. At \(\theta=\pi\), every frequency
\((2k+1)\pi/\log q\) is doubled per reciprocal pair.

Let \(m_0\) and \(m_\gamma\) be the actual zero multiplicities of
\(X_C\) at \(0\) and \(i\gamma\). The periodic frequency towers give
\(\sum_{\gamma>0}m_\gamma/\gamma^2<\infty\), hence the normally
convergent paired product

\[
X_C(z)
=Cz^{m_0}\prod_{\gamma>0}
\left(1+\frac{z^2}{\gamma^2}\right)^{m_\gamma}.
\]

A Hadamard quotient could only contribute \(e^{az+b}\); evenness forces
\(a=0\). Logarithmic differentiation gives

\[
H_C(x)
=\frac{m_0}{x}
+2\sum_{\gamma>0}\frac{m_\gamma}{x+\gamma^2},
\]

whose negative divided difference is the displayed Gram kernel.

```text
exact_map: PASS -- canonical Jacobian polarization -> positive Rosati identity pi^dagger*pi=[q] -> componentwise complex conjugation on Q[pi] -> pure normalized Frobenius roots -> paired canonical product -> Stieltjes Gram kernel -> positive diagonal congruence
counterfeit: P_-(u)=1+7u+9u^2 has the reciprocal equation and positive integral formal closed-point exponents in every degree but unequal normalized root moduli
counterfeit_failure: PASS -- its centered function has off-axis zeros and an Arb-certified negative order-two Loewner determinant, so reciprocity and positive Euler data do not replace Rosati positivity
success_delta: PASS -- C-0016 now has a primary-source-pinned proof, a fail-closed verifier, complete exceptional-multiplicity bookkeeping, and the sharpened domain lambda>0
failure_delta: PASS -- no finite-field argument transfers purity to the Riemann zeta function; a number-field analogue still requires an independently constructed positive global polarization
stop_condition: PASS -- a source mismatch, nilpotent Frobenius algebra, hidden additive logarithmic-derivative term, omitted theta=0 or theta=pi multiplicity, incorrect confluent value, or one nonpositive exact fixture minor would have refuted the claim
```

## Witness

The verifier reconstructs five honest curve numerators by exact finite-field
point counting and Newton identities. The strongest exceptional fixture is

\[
C:\ y^2=x^5-x\quad\text{over }\mathbf F_5,
\]

for which

\[
\#C(\mathbf F_{5^n})=(6,6,126,526)\quad(1\le n\le4),
\]

\[
P_C(u)=(1-5u^2)^2,\qquad
X_C(z)=4\sinh^2((\log5)z).
\]

It simultaneously exercises \(m_0=2\), both self-reciprocal angles, and
frequency multiplicity two. On this curve and four additional genus-one
and genus-two fixtures, every principal minor of the direct \(3\times3\)
\(M\) and \(Q\) matrices at

\[
\lambda\in\{1/4,3/4,5/4\}
\]

is strictly positive by 384-bit Arb enclosures.

## What survives

This is a function-field theorem, not evidence transferring purity to the
number-field case. A number-field analogue still requires a zero-independent
positive global polarization assembling every prime-power and archimedean
channel.

## Provenance

Source: Note 412, 2026-07-15. Audited and source-pinned on 2026-07-19
against J.S. Milne, *Abelian Varieties* v2.00 (2008), *Complex
Multiplication* v0.10 (2020), and *Jacobian Varieties* (2021). The audit
uses Theorem 11.1 over all finite extensions and the exact formal
power-sum derivation; neither the 2008 proof passage carrying an admitted
footnote nor the 2021 Corollary 11.4 polynomial-convention typo is used as
the numerator identity.
