On the structure of modules of vector valued modular forms

TL;DR

This paper explores the algebraic structure of modules of vector valued modular forms for representations of SL2(Z), providing bounds on generator weights and multiplicities, and proposing conjectures supported by computational and theoretical evidence.

Contribution

It introduces new bounds on the weights and multiplicities of generators of vector valued modular form modules, and investigates their structural properties.

Findings

Established bounds on weights of generators.

Provided bounds on multiplicities of weights.

Suggested a possible three-term multiplicity bound.

Abstract

If $\rho$ denotes a finite dimensional complex representation of $\textbf{SL}_2(\textbf{Z})$, then it is known that the module $M(\rho)$ of vector valued modular forms for $\rho$ is free and of finite rank over the ring $M$ of scalar modular forms of level one. This paper initiates a general study of the structure of $M(\rho)$. Among our results are absolute upper and lower bounds, depending only on the dimension of $\rho$, on the weights of generators for $M(\rho)$, as well as upper bounds on the multiplicities of weights of generators of $M(\rho)$. We provide evidence, both computational and theoretical, that a stronger three-term multiplicity bound might hold. An important step in establishing the multiplicity bounds is to show that there exists a free-basis for $M(\rho)$ in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series $E_6$ of weight six.