Jacobi Forms of Critical Weight and Weil Representations

TL;DR

This paper explores the deep connection between Jacobi forms of critical weight and Weil representations, providing theoretical insights, proving vanishing theorems, and confirming a conjecture on Jacobi forms of weight one.

Contribution

It offers a detailed analysis of the relation between critical weight Jacobi forms and Weil representations, including new proofs and applications.

Findings

Established the link between critical weight Jacobi forms and Weil representations.

Proved vanishing theorems for certain Jacobi forms.

Confirmed a conjecture on Jacobi forms of weight one with character.

Abstract

Jacobi forms can be considered as vector valued modular forms, and Jacobi forms of critical weight correspond to vector valued modular forms of weight $\frac12$. Since the only modular forms of weight $\frac12$ on congruence subgroups of $\SL$ are theta series the theory of Jacobi forms of critical weight is intimately related to the theory of Weil representations of finite quadratic modules. This article explains this relation in detail, gives an account of various facts about Weil representations which are useful in this context, and it gives some applications of the theory developed herein by proving various vanishing theorems and by proving a conjecture on Jacobi forms of weight one on $\SL$ with character.