The Picard group of the moduli space of curves with level structures

Abstract

For $4\nmid L$ and $g$ large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level $L$ structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces, and we calculate the second integral cohomology group of the level $L$ subgroup of the mapping class group. (In a previous paper, the author determined this rationally.) This entails calculating the abelianization of the level $L$ subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of ${Sp}_{2g}(\mathbb{Z}/L)$ with coefficients in the adjoint representation.