Cohomology of the Universal Abelian Surface with Applications to Arithmetic Statistics

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 $\pi$ 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 $\mathcal X_2^{\operatorname{Sym}(n)} \to \mathcal A_2$ for all $n \geq 1$, where the fiber over a point corresponding to an abelian surface $A$ is $A^n$ and $\operatorname{Sym}^n A$ respectively. We describe how to compute the $\ell$-adic cohomology of $\mathcal X_2^n$ and $\mathcal X_2^{\operatorname{Sym}(n)}$ 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 $\mathbf F_q$-points.