Taylor expansions of Jacobi forms and linear relations among theta series
Tom 590 / 2023
Dissertationes Mathematicae 590 (2023), 66 pp.
MSC: Primary 11F50; Secondary 11F37, 11F11, 11F27
DOI: 10.4064/dm880-12-2023
Opublikowany online: 3 January 2024
Streszczenie
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, 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 a complete SageMath code for the $D_4$ lattice.
