Cohomology of the boundary of Siegel modular varieties of degree two, with applications

J. William Hoffman¹ ; Steven H. Weintraub²

¹ Department of Mathematics Louisiana State University Baton Rouge, LA 70803, U.S.A.
² Department of Mathematics Lehigh University Bethlehem, PA 18015, U.S.A.

Fundamenta Mathematicae, Tome 178 (2003) no. 1, pp. 1-47.

Résumé

Let ${\mathcal{A}}_2(n)={\Gamma }_2(n)\mathrm{\setminus }{\mathfrak{S}}_2$ be the quotient of Siegel's space of degree 2 by the principal congruence subgroup of level $n$ in $\mathbf{S}\mathbf{p}(4,\mathbb{Z})$. 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 $\mathrm{H}(\partial {\mathcal{A}}_2(n{)}^{\ast })$ as a module over the finite group ${\Gamma }_2(1)/{\Gamma }_2(n)$. As an application we compute the cohomology of the arithmetic group ${\Gamma }_2(3)$.

Détail

DOI : 10.4064/fm178-1-1

Mots-clés : mathcal vargamma backslash mathfrak quotient siegels space degree principal congruence subgroup level mathbb moduli space principally polarized abelian surfaces level structure mathcal ast denote igusa compactification space partial mathcal ast mathcal ast mathcal its boundary divisor normal crossings main result paper determination partial mathcal ast module finite group vargamma vargamma application compute cohomology arithmetic group vargamma