Vector-Valued Modular Forms and the Gauss Map

We use the gradients of theta functions at odd two-torsion points --- thought of as vector-valued modular forms --- to construct holomorphic differential forms on the moduli space of principally polarized abelian varieties, and to characterize the locus of decomposable abelian varieties in terms of the Gauss images of two-torsion points.

2010 Mathematics Subject Classification: Primary:11F46 and secondary:14K25, 14K10

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