Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

TL;DR

This paper extends the Petersson inner product to non-cuspidal Jacobi forms, enabling better analysis of Fourier coefficients and applications to quadratic forms and Siegel modular forms.

Contribution

It introduces a generalized Petersson inner product for non-cuspidal Jacobi forms and explores its implications for growth estimates and form determination.

Findings

Enhanced understanding of Fourier Jacobi coefficient growth

Proved Siegel modular forms are determined by fundamental Fourier coefficients

Applications to quadratic form representation numbers

Abstract

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms, and as a by-product we also show that \textit{any} Siegel modular form of degree $2$ is determined by its `fundamental' Fourier coefficients.