Modules of Jacobi forms of degree two of small levels

TL;DR

This paper explicitly describes the structure of modules of degree two Jacobi forms of small levels, extending known results for full modular groups and specific levels, providing a clearer understanding of their algebraic structure.

Contribution

It provides explicit module structure theorems for Jacobi forms of degree two and small levels, merging previous results for scalar and vector valued forms.

Findings

Explicit module structures for levels 2, 3, 4

Unified approach combining previous results

Simplified descriptions of Jacobi form modules

Abstract

The purpose of this paper is to describe explicitly the modules of (Siegel-)Jacobi forms of degree two of index one of any scalar valued weight with respect to some congruence subgroups of small levels $N\leq 4$. Such a structure for the full Siegel modular group as a module over scalar valued Siegel modular forms of even weight has been explicitly given by T.~Ibukiyama. There we used an explicit structure theorem of rings of scalar valued Siegel modular forms by Igusa and that of vector valued Siegel modular forms of weight $\det^k\, \Sym^2$ by T. Satoh and T. Ibukiyama. On the other hand, for levels $N=2$, $3$, $4$, ring structures of scalar valued case have been also known by H. Aoki and T. Ibukiyama and $\det^k \Sym^2$-valued case by H. Aoki. In this paper, by merging these results, we give the same sort of simple structure theorems on modules of Jacobi forms of degree of two of index one for level $2$, $3$ and $4$.