Modified Jacobi forms of index zero (II)

TL;DR

This paper investigates the structure of modified Jacobi forms of negative weight and index zero, demonstrating that certain subspaces are finite-dimensional by analyzing Fourier coefficients.

Contribution

It introduces a subspace hierarchy for modified Jacobi forms and proves the finite-dimensionality of these subspaces.

Findings

Finite-dimensionality of subspaces $J_k^m$ for modified Jacobi forms.

Establishment of a union decomposition $J_k = igcup_{m=1}^ J_k^m$.

Relation between Fourier coefficients and subspace structure.

Abstract

For a negative integer $k$ let $J_k$ be the space of modified Jacobi forms of weight $k$ and index 0 on $\mathrm{SL}_2(\mathbb{Z})$. For each positive integer $m$ we consider certain subspace $J_k^{m}$ of $J_k$ which satisfies $J_k=\cup_{m=1}^\infty J_k^m$. By observing a relation between coefficients of the Fourier development of a modified Jacobi form we show that $J_k^m$ is finite-dimensional.