On the structure of modules of vector-valued modular forms

If ρ$$\rho $$ denotes a finite-dimensional complex representation of SL2(Z)$$\mathbf {SL}_{2}(\mathbf {Z})$$, then it is known that the module M(ρ)$$M(\rho )$$ of vector-valued modular forms for ρ$$\rho $$ 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(ρ)$$M(\rho )$$. Among our results are absolute upper and lower bounds, depending only on the dimension of ρ$$\rho $$, on the weights of generators for M(ρ)$$M(\rho )$$, as well as upper bounds on the multiplicities of weights of generators of M(ρ)$$M(\rho )$$. 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(ρ)$$M(\rho )$$ in which the matrix of the modular derivative operator does not contain any copies of the Eisenstein series E6$$E_6$$ of weight six.