Vector-valued modular forms and Poincaré series

Abstract

We initiate a general theory of vector-valued modular forms associated to a finite-dimensional representation of SL(2, Z). We introduce vector-valued Poincar series and Eisenstein series and a version of the Petersson inner product, and establish analogs of basic results from the classical theory of modular forms concerning these objects, at least if the weight is large enough. In particular, we show that the space of entire vector-valued modular forms of weight k associated to is a finite-dimensional vector space which, for large enough k, is nonzero and spanned by Poincar series. We show that Hecke's estimate an = O(n k-1 ) continues to apply to the Fourier coefficients of component functions of entire vector-valued modular forms associated to for large enough k.