Tautological modular forms of level two and degree two

TL;DR

This paper develops a method to construct all vector-valued Siegel modular forms of level two and degree two using divisors on the Hodge bundle and invariant theory, linking them to genus two curves.

Contribution

It introduces a novel approach combining divisors on the Hodge bundle and invariant theory to explicitly construct all such modular forms.

Findings

Constructed all vector-valued Siegel modular forms of level two and degree two.

Connected modular forms to the moduli space of genus two curves.

Provided explicit descriptions in terms of basic modular forms.

Abstract

We show how to use divisors on the projectivized Hodge bundle to construct special vector-valued modular forms and then apply invariant theory to construct all vector-valued Siegel modular forms of level two and degree two. Thus we construct all modular forms in terms of certain basic modular forms that are intimately connected to the moduli of curves of genus two.