The Siegel Modular Variety of Degree Two and Level Three 3

Let A2(n) denote the quotient of the Siegel upper half space of degree two by ?2(n), the principal congruence subgroup of level n in Sp(4; Z). A2(n) is the moduli space of principally polarized abelian varieties of dimension two with a level n structure, and has a compacti-cation A2(n) rst constructed by Igusa. When n 3 this is a smooth projective algebraic variety of dimension three. In this work we analyze the topology of A2(3) and the open subset A2(3). In this way we obtain the rational cohomology ring of ?2(3). The key is that one has an explicit description of A2(3) : it is the resolution of the 45 nodes on a projective quartic threefold whose equation was rst written down about 100 years ago by H. Burkhardt. We are able to compute the zeta function of this variety reduced modulo certain primes.