The Picard group of the moduli space of curves with level structures - [pdf] [ps]
Duke Math. J. 161 (2012), no. 4, 623–674.
Computer code used in paper here.
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.