**MSC:**- 11F46 Siegel modular groups and their modular and automorphic forms
- 14G35 Modular and Shimura varieties, See also {11F41, 11F46,

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}