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 Sp_(2g) (Z/ L) with coefficients in the adjoint representation.
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