VECTOR-VALUED MODULAR FORMS AND LINEAR DIFFERENTIAL OPERATORS

Abstract

We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, we prove that the inequality k ≥ 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0. For any p ≥ 2, we show how to construct large numbers of representations of Γ for which k = 2 - p. The key idea is to consider representations of Γ on spaces of solutions of certain linear differential equations whose coefficients are modular forms.