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

TL;DR

This paper computes the integral Picard groups of moduli spaces of curves with level structures, revealing divisibility properties of line bundles and the structure of related cohomology groups for large genus and specific level conditions.

Contribution

It provides new calculations of the Picard groups and second cohomology groups for moduli spaces with level structures, extending previous rational results to integral ones and analyzing the abelianization of level subgroups.

Findings

Determined the integral Picard groups for large genus and specific levels.

Calculated the second integral cohomology group of the level L subgroup of the mapping class group.

Analyzed the abelianization of the level L subgroup and the first homology of the mod L symplectic group.

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 the mod $L$ symplectic group with coefficients in the adjoint representation.