Cohomological Arithmetic Statistics for Principally Polarized Abelian Varieties over Finite Fields

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 $g\rightarrow\infty$.