Basic vector valued Siegel modular forms of genus two

TL;DR

This paper investigates the structure of vector valued Siegel modular forms of genus two over certain groups, using theta functions and derivatives to provide a complete description for specific cases.

Contribution

It offers a detailed analysis of modules of vector valued Siegel modular forms for particular groups, revealing their structure with new explicit descriptions.

Findings

Complete structure of modules for $\Gamma[4,8]$ and $\Gamma[2,4]$

Use of theta functions and derivatives as main tools

Focus on standard and Sym^2 representations

Abstract

We study over rings of scalar valued Siegel modular forms. modules of vector valued modular forms of degree two. For the two simplest representations, standard and Sym^2, appears rather natural consider the cases of the group $\Gamma[4,8] $ and $\Gamma[2,4].$ In these case we give the complete structure of the modules . The main tools are Theta functions and their derivatives evaluated at $z=0$