On the structure of modules of vector-valued modular forms

If (rho ) denotes a finite-dimensional complex representation of (mathbf {SL}_{2}(mathbf {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.