Cohomology of the Universal Abelian Surface with Applications to Arithmetic Statistics

Abstract

Abstract The moduli stack $\mathcal{A}_2$ of principally polarized abelian surfaces comes equipped with the universal abelian surface $\pi: \mathcal{X}_2 \to \mathcal{A}_2$. The fiber of π over a point corresponding to an abelian surface A in $\mathcal{A}_2$ is A itself. We determine the $\ell$-adic cohomology of $\mathcal{X}_2$ as a Galois representation. Similarly, we consider the bundles $\mathcal{X}_2^n \to \mathcal{A}_2$ and $\mathcal{X}_2^{\textrm{Sym}(n)} \to \mathcal{A}_2$ for all $n \geq 1$, where the fiber over a point corresponding to an abelian surface A is An and $\textrm{Sym}^n A$, respectively. We describe how to compute the $\ell$-adic cohomology of $\mathcal{X}_2^n$ and $\mathcal{X}_2^{\textrm{Sym}(n)}$ and explicitly calculate it in low degrees for all n and in all degrees for n = 2. These results yield new information regarding the arithmetic statistics on abelian surfaces, including an exact calculation of the expected value and variance as well as asymptotics for higher moments of the number of Fq-points.