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

Abstract

For 4 | 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(Z{double-struck}/L) with coefficients in the adjoint representation. © 2012.