Dissertationes mathematicae

Abstract

We study Taylor expansions of Jacobi forms of lattice index. As the main result, we give an embedding from a certain space of such forms, whether scalar-valued or vector-valued, integralweight 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 D4 lattice and A3 lattice are obtained, along with relations among the third powers of series associated with the A2 lattice. We present a complete SageMath code for the D4 lattice.