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Taylor expansions of Jacobi forms and linear relations among theta series

Volume 590 / 2023

Xiao-Jie Zhu Dissertationes Mathematicae 590 (2023), 66 pp. MSC: Primary 11F50; Secondary 11F37, 11F11, 11F27 DOI: 10.4064/dm880-12-2023 Published online: 3 January 2024


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.


  • Xiao-Jie ZhuSchool of Mathematical Sciences
    East China Normal University
    200241 Shanghai, P.R. China

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