Constructions of vector-valued modular forms of rank four and level one

TL;DR

This paper investigates the structure of vector-valued modular forms of rank four and level one, classifying their isomorphism types, deriving associated differential equations, and solving them explicitly in special cases.

Contribution

It provides a classification of the isomorphism types of rank four modular form spaces and explicit solutions for differential equations in specific representation cases.

Findings

Two possible isomorphism types identified for the space of modular forms.

Differential equations for minimal weight forms derived and solved explicitly in certain cases.

Methods to construct the entire graded module of modular forms from minimal weight forms.

Abstract

This paper studies modular forms of rank four and level one. There are two possiblities for the isomorphism type of the space of modular forms that can arise from an irreducible representation of the modular group of rank four, and we describe when each case occurs for general choices of exponents for the T -matrix. In the remaining sections we describe how to write down corresponding differential equations satisfied by minimal weight forms, and how to use these minimal weight forms to describe the entire graded module of holomorphic modular forms. Unfortunately the differential equations that arise can only be solved recursively in general. We conclude the paper by studying the cases of tensor products of two-dimensional representations, symmetric cubes of two-dimensional representations, and inductions of two-dimensional representations of the subgroup of the modular group of index two. In these cases the differential equations satisfied by minimal weight forms can be solved exactly.