Dimension formulas for spaces of vector-valued Siegel modular forms of degree two and level two

TL;DR

This paper determines the detailed structure of vector-valued Siegel modular forms of degree two and level two by analyzing cohomology and Euler characteristics, revealing their decomposition under symmetry group actions.

Contribution

It provides the first explicit isotypical decomposition of these modular form spaces using cohomological methods and Euler characteristic computations.

Findings

Decomposition of modular form spaces under symmetric group actions

Explicit formulas for cohomology of local systems on moduli spaces

Identification of isotypical components in the modular forms

Abstract

Using a description of the cohomology of local systems on the moduli space of abelian surfaces with a full level two structure, together with a computation of Euler characteristics we find the isotypical decomposition, under the symmetric group on 6 letters, of spaces of vector-valued Siegel modular forms of degree two and level two.