Theta Series Associated with the Weil-Schroedinger Representation

TL;DR

This paper introduces the Weil-Schroedinger representation of the Jacobi group and demonstrates that theta series linked to this representation are Jacobi forms under specific arithmetic subgroups, enhancing understanding of their transformation properties.

Contribution

It defines the Weil-Schroedinger representation and proves the associated theta series are Jacobi forms, extending the classical Weil representation framework.

Findings

Theta series associated with the Weil-Schroedinger representation are Jacobi forms.

The paper establishes transformation properties under a specific arithmetic subgroup.

Provides a new perspective on the role of the Weil representation in modular forms.

Abstract

The Weil representation discovered by Andre Weil plays an important role in the study of the tranformation properties of theta series. In this paper, we define the Weil-Schroedinger representation of the Jacobi group and prove that the theta series associated with the Weil-Schroedinger representation is a Jacobi form with respect to a suitable arithmetic subgroup of the Jacobi modular group.