Cohomology of the boundary of Siegel modular varieties of degree two, with applications
Volume 178 / 2003
Abstract
Let $\mathcal A_{2}(n) = \varGamma _{2}(n)\backslash {\mathfrak S}_{2}$ be the quotient of Siegel's space of degree 2 by the principal congruence subgroup of level $n$ in ${\bf Sp}(4, \mathbb Z)$. This is the moduli space of principally polarized abelian surfaces with a level $n$ structure. Let $\mathcal A_{2}(n)^{\ast}$ denote the Igusa compactification of this space, and $\partial\mathcal A_2(n)^{\ast} = \mathcal A_2(n)^{\ast} - \mathcal A_2(n)$ its “boundary”. This is a divisor with normal crossings. The main result of this paper is the determination of ${\rm H}(\partial\mathcal A_2(n)^{\ast})$ as a module over the finite group $\varGamma _{2}(1) / \varGamma _{2}(n)$. As an application we compute the cohomology of the arithmetic group $\varGamma _{2}(3)$.
