Papers With

← All papers
First page of Point counts of abelian varieties over finite fields determining their zeta function

Point counts of abelian varieties over finite fields determining their zeta function

Shiva Chidambaram, Timo Keller

math.NT Jun 27, 2026 · v2 math.AG
formalization mathlib number-theory
Read PDF arXiv abstract Code
TL;DR
The recovery argument (Sections 3–6) is formalized in Lean 4 with Mathlib, with two sorries for classical identities missing from Mathlib.
Abstract
Let $A$ be an abelian variety of dimension $g$ over a finite field $\mathbf{F}_q$. We show that if $q$ is sufficiently large relative to $g$, the $g$ point counts $\#A(\mathbf{F}_{q^i})$ for $1 \leq i \leq g$ determine the zeta function of $A$, equivalently the characteristic polynomial of its Frobenius endomorphism, and hence the isogeny class of $A$. This count is best possible for $g=2$ and $g=4$, but not in general: for $g=3$ two point counts already determine the zeta function, whereas a single count never does. The proof combines the functional equation of the $L$-polynomial with Newton's identities and an inductive error analysis that controls the power sums of the inverse Frobenius eigenvalues with enough precision to recover them, as integers, by rounding.
Problem

For an abelian variety A of dimension g over F_q, the question is how many point counts #A(F_{q^i}) are needed to determine its zeta function (equivalently its L-polynomial and isogeny class). Kedlaya showed that 2g counts suffice for large q.

Approach

The proof combines the functional equation of the L-polynomial with Newton's identities. An inductive error analysis bounds the power sums of the inverse Frobenius eigenvalues precisely enough that they can be recovered exactly as integers by rounding. The argument treats the ranges i ≤ g/2 and g/2 < i ≤ g separately. The recovery argument is formalized in Lean 4 on top of Mathlib, apart from two sorries: infinite Möbius inversion and the explicit Newton–Girard formula, both absent from Mathlib.

Results

If q > (16g^3 p(2g))^{2g+2}, the g point counts #A(F_{q^i}), 1 ≤ i ≤ g, determine the zeta function. This count is optimal for g=2 and g=4. For g=3, two point counts suffice, and a single count never does.