On the Fourier coefficients of 2-dimensional vector-valued modular forms

Abstract

Let ρ : S L ( 2 , Z ) → G L ( 2 , C ) \rho : SL(2, \mathbb {Z}) \rightarrow GL(2, \mathbb {C}) be an irreducible representation of the modular group such that ρ ( T ) \rho (T) has finite order N N . We study holomorphic vector-valued modular forms F ( τ ) F(\tau ) of integral weight associated to ρ \rho which have rational Fourier coefficients. (These span the complex space of all integral weight vector-valued modular forms associated to ρ \rho .) As a special case of the main theorem, we prove that if N N does not divide 120 120 , then every nonzero F ( τ ) F(\tau ) has Fourier coefficients with unbounded denominators.