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

Abstract

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.

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

Cited by in corpus (14)

• Representation theory and homological stability
• Representation stability and finite linear groups
• Classical Codes and Chiral CFTs at Higher Genus
• Mapping class groups of highly connected $(4k+2)$-manifolds
• Differential geometric invariants for time-reversal symmetric Bloch-bundles: the "Real" case
• Geometric quantization and the metric dependence of the self-dual field theory
• The anomaly line bundle of the self-dual field theory
• The Picard group of the moduli space of r-Spin Riemann surfaces
• The global gravitational anomaly of the self-dual field theory
• Cohomology of symplectic groups and Meyer's signature theorem
• From 5d Flat Connections to 4d Fluxes (the Art of Slicing the Cone)
• Monodromy of Codimension-One Sub-Families of Universal Curves
• Two functions on Sp(g,R)
• Deformation rings and images of Galois representations