J. William Hoffman, Steven H. Weintraub
The Siegel Modular Variety of Degree Two and Level Three
Preprint series: Mathematica Gottingensis

MSC:

11F46 Siegel modular groups and their modular and automorphic forms

14G35 Modular and Shimura varieties

Abstract: \begin{abstract}
Let $\A{2}{n}{}$ denote the quotient of the Siegel upper half space
of degree two by $\Gam{2}{n}$, the principal congruence subgroup of
level $n$ in $\sym{4}{\Z}$. $\A{2}{n}{}$
is the moduli space of principally polarized abelian varieties of
dimension two with a level
$n$ structure, and has a compactification $\A{2}{n}{\ast}$ first
constructed by Igusa. When
$n \ge 3$ this is a smooth projective algebraic variety of
dimension three.\par
In this work we analyze the topology of $\A{2}{3}{\ast}$ and the
open subset $\A{2}{3}{}$.
In this way we obtain the rational cohomology ring of $\Gam{2}{3}$.
The key is that one has an explicit description of $\A{2}{3}{\ast}$
: 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.
\end{abstract}

Keywords: Siegel modular variety, Igusa compactification, Burkhardt quartic