Note on Hermitian Jacobi Forms

TL;DR

This paper investigates Hermitian Jacobi forms, comparing them with classical Jacobi forms, establishing bounds on their vanishing order, and analyzing their algebraic structure and independence of generators.

Contribution

It introduces an embedding of Hermitian Jacobi forms into classical Jacobi forms, derives bounds on vanishing orders, and proves algebraic independence of generators for index 1.

Findings

Derived upper bounds for vanishing order at the origin.

Computed the rank of Hermitian Jacobi forms as a module over elliptic modular forms.

Proved algebraic independence of generators for index 1.

Abstract

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the origin is obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.