Cohomology of the Boundary of Siegel Modular Varieties of Degree Two, with Applications

Let A2(n) = Γ2(n)\S2 be the quotient of Siegel’s space of degree 2 by the principal congruence subgroup of level n in Sp(4, Z). This is the moduli space of principally polarized abelian surfaces with a level n structure. Let A2(n) ∗ denote the Igusa compactification of this space, and ∂A2(n) ∗ = A2(n) ∗−A2(n) its “boundary”. This is a divisor with normal crossings. The main result of this paper is the determination of H∗(∂A2(n) ∗) as a module over the finite group Γ2(1)/Γ2(n). As an application we compute the cohomology of the arithmetic group Γ2(3).