2-Dimensional vector-valued modular forms

Abstract

We study the graded space ℋ(ρ) of holomorphic vector-valued modular forms of integral weight associated to a 2-dimensional irreducible representation ρ of SL(2,ℤ). When ρ(T) is unitary, a complete description of ℋ(ρ) is given: the Poincaré series is calculated and it is shown that ℋ(ρ) is a free module of rank 2 over the ring of (classical) holomorphic modular forms on SL(2,ℤ).