Covariants of binary sextics and modular forms of degree 2 with character

TL;DR

This paper explores the use of covariants of binary sextics to understand the structure of Siegel modular forms of degree 2 with character, linking algebraic covariants to modular form properties.

Contribution

It introduces a novel approach connecting covariants of binary sextics to the structure of modular forms with character, providing explicit descriptions and vanishing order formulas.

Findings

Describes the module structure of Siegel modular forms using covariants.

Provides formulas for the order of vanishing along specific loci.

Establishes a link between algebraic covariants and modular form properties.

Abstract

We use covariants of binary sextics to describe the structure of modules of scalar-valued or vector-valued Siegel modular forms of degree 2 with character, over the ring of scalar-valued Siegel modular forms of even weight. For a modular form defined by a covariant we express the order of vanishing along the locus of products of elliptic curves in terms of the covariant.