Generating Picard modular forms by means of invariant theory

TL;DR

This paper develops a method to generate all vector-valued Picard modular forms for discriminant -3 using invariant theory, linking modular forms to binary forms and their bi-covariants.

Contribution

It introduces a novel approach to construct Picard modular forms via bi-covariants of binary forms, connecting invariant theory with modular form generation.

Findings

All vector-valued Picard modular forms can be generated from bi-covariants.

The universal binary forms relate to specific modular forms and Eisenstein series.

The approach provides explicit constructions for modular forms of certain weights.

Abstract

We use the description of the Picard modular surface for discriminant $-3$ as a moduli space of curves of genus $3$ to generate all vector-valued Picard modular forms from bi-covariants for the action of ${GL}_2$ on the space of pairs of binary forms of bidegree $(4,1)$. The universal binary forms of degree $4$ and $1$ correspond to a meromorphic modular form of weight $(4,-2)$ and a holomorphic Eisenstein series of weight $(1,1)$.