  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.
