The Picard group of the moduli space of curves with level structures
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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 the mod L symplectic group with coefficients in the adjoint representation.
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Putman, Andrew. "The Picard group of the moduli space of curves with level structures." Duke Mathematical Journal, 161, no. 4 (2012) 623-674. http://dx.doi.org/10.1215/00127094-1548362.