Vector Valued Siegel Modular Forms for \Gamma_2[2,4] and Sym^2

TL;DR

This paper establishes structure theorems for vector valued Siegel modular forms associated with a specific subgroup, linking them to rational tensors and theta series, and showing they are generated by Rankin-Cohen brackets.

Contribution

It introduces two structure theorems for vector valued Siegel modular forms for rac12;_2[2,4], connecting them to rational tensors and theta series, and demonstrating generation by Rankin-Cohen brackets.

Findings

Modules are generated by Rankin-Cohen brackets of theta series.

Identifies modular forms with rational tensors with manageable poles.

Provides explicit structure theorems for these modular forms.

Abstract

We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup \Gamma_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular forms with rational tensors with easily handleable poles on P^3\C. It follows that the observed modules of modular forms are generated by the Rankin-Cohen brackets of the four theta series of the second kind.