Some aspects of Hermitian Jacobi forms

TL;DR

This paper develops a differential operator for Hermitian Jacobi forms, explores its properties, and constructs new forms via lifts and tensor products, advancing understanding of their structure and Fourier coefficients.

Contribution

It introduces a heat operator on Hermitian Jacobi forms, constructs lifts from elliptic cusp forms, and embeds subspaces into modular forms, providing new tools and insights.

Findings

Defined a differential (heat) operator for Hermitian Jacobi forms

Constructed Hermitian Jacobi forms from tensor products and differentiation

Embedded subspaces into sums of modular forms

Abstract

We introduce a certain differential (heat) operator on the space of Hermitian Jacobi forms of degree 1, show it's commutation with certain Hecke operators and use it to construct a lift of elliptic cusp forms to Hermitian Jacobi cusp forms. We construct Hermitian Jacobi forms as the image of the tensor product of two copies of Jacobi forms and also from differentiation of the variables. We determine the number of Fourier coefficients that determine a Hermitian Jacobi form and use it to embed a certain subspace of Hermitian Jacobi forms into a direct sum of modular forms for the full modular group.