The Siegel modular variety of degree two and level three

Abstract:

Let $\mathcal {A}_{2}(n)$ denote the quotient of the Siegel upper half space of degree two by $\Gamma _{2}(n)$, the principal congruence subgroup of level $n$ in $\mathbf {Sp}(4,\mathbf {Z})$. $\mathcal {A}_{2}(n)$ is the moduli space of principally polarized abelian varieties of dimension two with a level $n$ structure, and has a compactification $\mathcal {A}_{2}(n)^{\ast }$ first constructed by Igusa. When $n\ge 3$ this is a smooth projective algebraic variety of dimension three. In this work we analyze the topology of $\mathcal {A}_{2}(3)^{\ast }$ and the open subset $\mathcal {A}_{2}(3)$. In this way we obtain the rational cohomology ring of $\Gamma _{2}(3)$. The key is that one has an explicit description of $\mathcal {A}_{2}(3)^{\ast }$: it is the resolution of the 45 nodes on a projective quartic threefold whose equation was first written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.