On the Space of Slow Growing Weak Jacobi Forms

TL;DR

This paper investigates the space of weak Jacobi forms of weight 0 and index m that produce slow-growing Fourier coefficients upon lifting to Siegel paramodular forms, providing evidence for their existence at all indices.

Contribution

It characterizes the space of weak Jacobi forms associated with slow growth and offers analytic and numerical evidence supporting their universal existence across all indices.

Findings

Existence of slow growing weak Jacobi forms for all indices

Analytic and numerical support for the conjecture

Characterization of the space leading to slow growth

Abstract

Weak Jacobi forms of weight $0$ and index $m$ can be exponentially lifted to meromorphic Siegel paramodular forms. It was recently observed that the Fourier coefficients of such lifts are then either fast growing or slow growing. In this note we investigate the space of weak Jacobi forms that lead to slow growth. We provide analytic and numerical evidence for the conjecture that there are such slow growing forms for any index $m$.