Vector-valued modular forms and the Gauss map

TL;DR

This paper explores the use of vector-valued modular forms derived from theta function gradients to construct differential forms on moduli spaces and characterize decomposable abelian varieties via Gauss images.

Contribution

It introduces a novel approach linking theta function gradients to the geometry of abelian varieties and their moduli spaces.

Findings

Constructs holomorphic differential forms on moduli spaces

Characterizes decomposable abelian varieties using Gauss images

Establishes a new connection between modular forms and algebraic geometry

Abstract

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.