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 …