On the structure of modules of vector valued modular forms

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.