Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields

TL;DR

This paper explores the cohomological properties of moduli spaces of principally polarized Abelian varieties over finite fields, linking these to point count distributions and proposing asymptotic conjectures as the dimension grows.

Contribution

It computes cohomology and Euler characteristics for these moduli spaces in low dimensions and formulates conjectures on their asymptotic behavior for large dimensions.

Findings

Cohomology computations for g=1, 2, 3 cases.

Identification of polynomial ranges for point counts.

Conjecture on asymptotic point count behavior as g→∞.

Abstract

There is a natural probability measure on the set of isomorphism classes of principally polarized Abelian varieties of dimension $g$ over $\mathbb{F}_q$, weighted by the number of automorphisms. The distributions of the number of $\mathbb{F}_q$-rational points are related to the cohomology of fiber powers of the universal family of principally polarized Abelian varieties. To that end we compute the cohomology $H^i(\mathcal{X}^{\times n}_g,\mathbb{Q}_\ell)$ for $g=1$ using results of Eichler-Shimura and for $g=2$ using results of Lee-Weintraub and Petersen, and we compute the compactly supported Euler characteristics $e_\mathrm{c}(\mathcal{X}^{\times n}_g,\mathbb{Q}_\ell)$ for $g=3$ using results of Hain and conjectures of Bergstr\"om-Faber-van der Geer. In each of these cases we identify the range in which the point counts $\#\mathcal{X}^{\times n}_g(\mathbb{F}_q)$ are polynomial in $q$. Using results of Borel and Grushevsky-Hulek-Tommasi on cohomological stability, we adapt arguments of Achter-Erman-Kedlaya-Wood-Zureick-Brown to pose a conjecture about the asymptotics of the point counts $\#\mathcal{X}^{\times n}_g(\mathbb{F}_q)$ in the limit $g\rightarrow\infty$.