Structure of the module of vector-valued modular forms

Abstract

Let V be a representation of the modular group Γ of dimension p. We show that the Z-graded space H(V ) of holomorphic vector-valued modular forms associated to V is a free module of rank p over the algebra M of classical holomorphic modular forms. We study the nature of H considered as a functor from Γ-modules to graded M-lattices and give some applications, including the calculation of the Hilbert-Poincaré series of H(V ) in some cases.

On the structure of modules of vector-valued modular forms

“…[8] for more details). This shows that at the level of multisets, the set of weights for ρ is contained in the union of the corresponding multisets for ρ 1 and ρ 2 .…”

“…One knows ( [9], [8]) that M(ρ) is a Z-graded left R-module. Elements of M act by multiplication and D acts via the obvious extension of (4) to vvmfs of weight k. It is the exploitation of this fact that underlies the results in the present Section.…”

“…If ρ is a finite-dimensional complex representation of SL 2 (Z) of dimension d, then the module M(ρ) of vector valued modular forms for ρ is known [8], [2] to be free of rank d over the ring M of classical scalar modular forms of level one. A basic problem about M(ρ) is then to determine the weights of a generating set of modular forms in this module.…”

Three-dimensional imprimitive representations of the modular group and their associated modular forms

“…Section 4.3 of [6] explains how generalized hypergeometric series and the freemodule theorem of [15] allow one to describe the module of vector-valued modular forms associated to an irreducible three-dimensional representation of Γ. The answer is expressed in terms of the exponents of the eigenvalues of ρ(T ).…”

“…The idea of the proof of Theorem 21 is to transfer this special result for F into a general result about vector-valued modular forms. The methods for doing this are essentially already in the literature ( [16], [15], [5]), though they are not stated in the form that we need here. We will therefore give some of the details.…”