On vector-valued Siegel modular forms of degree 2 and weight (j,2)

TL;DR

This paper formulates a conjecture on vector-valued Siegel modular forms of degree 2 and weight (j,2), constructs examples using covariants, and provides evidence through Fourier expansions, contributing to the understanding of these modular forms.

Contribution

It introduces a conjecture on such modular forms, constructs explicit examples via covariants, and offers computational evidence supporting the conjecture.

Findings

Construction of modular forms of weight (j,2) via covariants

Fourier expansions illustrating the approach's effectiveness

Proof that all forms of weight (j,1) vanish at level 2

Abstract

We formulate a conjecture that describes the vector-valued Siegel modular forms of degree 2 and level 2 of weight Sym^j det^2 and provide some evidence for it. We construct such modular forms of weight (j,2) via covariants of binary sextics and calculate their Fourier expansions illustrating the effectivity of the approach via covariants. Two appendices contain related results of Chenevier; in particular a proof of the fact that every modular form of degree 2 and level 2 and weight (j,1) vanishes.