Modules of Jacobi forms of degree two of small levels

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 $4$.