For and large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level 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 subgroup of the mapping class group. (In a previous paper, the author determined this rationally.) This entails calculating the abelianization of the level 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 with coefficients in the adjoint representation.
"The Picard group of the moduli space of curves with level structures." Duke Math. J. 161 (4) 623 - 674, 15 March 2012. https://doi.org/10.1215/00127094-1548362