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

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

2009-08-04

arxiv.org/abs/0908.0555v3

Mathematics

Algebraic Geometry

32 pages, 1 figure; substantial revision. To appear in Duke Math. J

Scientific paper

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.

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