Note on Hermitian Jacobi Forms

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.