二阶小水平的雅可比形式的模
Modules of Jacobi forms of degree two of small levels
AI总结：
本文通过合并已知结果，为小水平2、3、4的二阶雅可比形式的模结构提供简单结构定理。
摘要：
本文旨在明确描述任何标量值权重下，针对小水平$N\leq 4$的某些同余子群的二阶雅可比形式（索引为一）的模结构。对于完整的西格尔模形式群作为标量值西格尔模形式的偶数权重的模结构，已由T．Ibukiyama明确给出。我们利用Igusa关于标量值西格尔模形式环的显式结构定理以及T．Satoh和T．Ibukiyama关于权重$\det^k\, \Sym^2$的向量值西格尔模形式的结构定理。另一方面，对于水平$N=2$，$3$，$4$，标量值情况已由H．Aoki和T．Ibukiyama所知，而$\det^k \Sym^2$-valued情况由H．Aoki所知。在本文中，通过合并这些结果，我们为水平$2$，$3$和$4$的二阶雅可比形式（索引为一）的模给出类似的简单结构定理。
英文摘要：
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 $4$.
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