On the vanishing order of Jacobi forms at infinity

Jialin Li, Haowu Wang

Published 2025-06-19Version 1

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.