On the vanishing order of Jacobi forms at infinity

TL;DR

This paper establishes optimal bounds on the vanishing order of Jacobi forms at infinity, impacting the understanding of modular forms and their Fourier-Jacobi series.

Contribution

It introduces two types of upper bounds for Jacobi forms' vanishing order, including for lattice index forms, and derives implications for orthogonal modular forms.

Findings

Optimal upper bounds for classical Jacobi forms' vanishing order.

Lower bounds on the slope of orthogonal modular forms.

Finite rank of the module of symmetric formal Fourier--Jacobi series.

Abstract

In this paper, we establish two types of upper bounds on the vanishing order of Jacobi forms at infinity. The first type is for classical Jacobi forms, which is optimal in a certain sense. The second type is for Jacobi forms of lattice index. Based on this bound, we obtain a lower bound on the slope of orthogonal modular forms, and we prove that the module of symmetric formal Fourier--Jacobi series on $\mathrm{O}(m,2)$ has finite rank.