Taylor expansions of Jacobi forms and linear relations among theta series

TL;DR

This paper establishes an embedding of Jacobi forms into modular forms spaces and explores linear relations among theta series, providing new insights into lattice-based modular objects and their algebraic relations.

Contribution

It introduces a general embedding of Jacobi forms into modular forms spaces and analyzes linear relations among theta series of specific lattices.

Findings

Linear relations among second powers of theta series for D4 and A3 lattices.

Relations among third powers of theta series for A2 lattice.

Complete SageMath code for D4 lattice theta series.

Abstract

We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from certain space of such forms, whether scalar-valued or vector-valued, integral-weight or half-integral-weight, of any level, with any character, into a product of finitely many spaces of modular forms. As an application, we investigate linear relations among Jacobi theta series of lattice index. Many linear relations among the second powers of such theta series associated with the $D_4$ lattice and $A_3$ lattice are obtained, along with relations among the third powers of series associated with the $A_2$ lattice. We present the complete SageMath code for the $D_4$ lattice.