The Siegel modular variety of degree two and level three (Q2716139)

Let \(\Gamma_2(n)\) be the principal congruence subgroup of level \(n\) in \(\text{Sp}(4,\mathbb Z)\), and let \(\mathfrak S_2\) be the Siegel upper-half space of degree 2. Then the associated Siegel modular variety \(\mathcal A_2 (n) = \Gamma_2 (n) \backslash \mathfrak S_2\) is the modular space of principally polarized abelian varieties of dimension 2 with a level \(n\) structure. Toroidal compactifications of \(\mathcal A_2 (n)\) were first constructed by Igusa, and they are smooth projective algebraic varieties of dimension 3 when \(n \geq 3\). In this paper the authors analyze the topology of the quasi-projective variety \(\mathcal A_2 (3)\) as well as its toroidal compactification \(\mathcal A_2 (3)^\ast\) and obtain the rational cohomology ring of \(\Gamma_2 (3)\). The methods used involve the computation of the zeta function of the variety \(\mathcal A_2 (3)^\ast\) reduced modulo certain primes and the description of \(\mathcal A_2 (3)^\ast\) as the resolution of the 45 nodes on a projective quartic threefold.