Siegel theta series for indefinite quadratic forms

TL;DR

This paper generalizes the understanding of the modular transformation behavior of theta series from indefinite quadratic forms to Siegel theta series, extending prior elliptic modular form results.

Contribution

It introduces a generalization of Vignéras' differential equation criterion from elliptic to Siegel theta series for indefinite quadratic forms.

Findings

Extended the differential equation criterion to Siegel theta series.

Provided a theoretical framework for modularity of Siegel theta series.

Enhanced understanding of indefinite quadratic forms in higher dimensions.

Abstract

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vign\'eras, who deduced that solving a differential equation of second order serves as a criterion for modularity. In this paper, we will give a generalization of this result to Siegel theta series.