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Modules of Jacobi forms of degree two of small levels

Hiroki Aoki, Tomoyoshi Ibukiyama

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 $4$.

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 . 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 by T. Satoh and T. Ibukiyama. On the other hand, for levels , , , ring structures of scalar valued case have been also known by H. Aoki and T. Ibukiyama and -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 , and .
Paper Structure (19 sections, 21 theorems, 302 equations)

This paper contains 19 sections, 21 theorems, 302 equations.

Table of Contents

  1. Problems and theorems
  2. Review on modular forms and differential operators
  3. Siegel modular forms and Jacobi forms
  4. Differential operators on Siegel modular forms and Jacobi forms
  5. Relation between Jacobi forms and Siegel modular forms
  6. Witt operator and plus and minus part
  7. Theta constants and its transformation formula
  8. Proof for level $1$ and $2$
  9. Proof for level $3$
  10. Structure of $J_{*,1}^+(\Gamma_0(3)^{\psi})$
  11. Structure of $J_{*,1}^-(\Gamma_0(3)^{\psi})$
  12. Siegel modular forms of level $4$
  13. Graded rings of level $4$
  14. Proof for $\Gamma_0(4)^{\psi}$
  15. Proof for type I
  16. ...and 4 more sections

Key Result

Theorem 1.1

Each space $J_{*,1}^{I}(\Gamma)$ or $J_{*,1}^{II}(\Gamma)$ in the left column of the following table is a free module over $A^+(\Gamma)$ of rank $4$ and the weights of generators are given in the right column. All the generators are explicitly given, which will be explained later.

Theorems & Definitions (31)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3: Igusa igusagraded1 p.227
  • Lemma 2.4: Igusa igusagraded2 Lemma 4
  • Lemma 2.5
  • proof
  • ...and 21 more