number theory

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

arXiv:2606.28989

summary

The paper proves that for an abelian variety of dimension g over a sufficiently large finite field, the point counts over the first g extensions uniquely determine its zeta function and thus its isogeny class.

Abstract

Let $A$ be an abelian variety of dimension $g$ over a finite field $F_{q}$ . We show that if $q$ is sufficiently large relative to $g$ , the $g$ point counts $# A (F_{q^{i}})$ 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.

17 pages; comments welcome; Sage, Python, Lean code on https://github.com/TimoKellerMath/PointCountsAbelianVarieties

Topics & keywords

#abelian varieties #finite fields #zeta functions #point counts #isogeny classesFrobenius endomorphismcharacteristic polynomialL-polynomialNewton's identitiesSage codePython implementation