Cohomology of the Universal Abelian Surface with Applications to Arithmetic Statistics

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