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.
An algebraic classification is given for spaces of holomorphic vector-valued modular forms of arbitrary real weight and multiplier system, associated to irreducible, T -unitarizable representations of the full modular group, of dimension less than six. For representations of dimension less than four, it is shown that the associated space of vectorvalued modular forms is a cyclic module over a certain skew polynomial ring of differential operators. For dimensions four and five, a complete list of possible Hilbert-Poincaré series is given, using the fact that the space of vector-valued modular forms is a free module over the ring of classical modular forms for the full modular group. A mild restriction is then placed on the class of representation considered in these dimensions, and this again yields an explicit determination of the associated Hilbert-Poincaré series.
We prove that only a finite number of three-dimensional, irreducible representations of the modular group admit vector-valued modular forms with bounded denominators. This provides a verification, in the three-dimensional setting, of a conjecture concerning the Fourier coefficients of noncongruence modular forms, and reinforces the understanding from mathematical physics that when such a representation arises in rational conformal field theory, its kernel should be a congruence subgroup of the modular group.2000 Mathematics Subject Classification. 11F30, 11F99.
We classify the three-dimensional representations of the modular group that are reducible but indecomposable, and their associated spaces of holomorphic vector-valued modular forms. We then demonstrate how such representations may be employed to compute periods of modular curves. This technique obviates the use of Hecke operators, and therefore provides a method for studying noncongruence modular curves as well as congruence.
Using the language of vertex operator algebras (VOAs) and vector-valued modular forms we study the modular group representations and spaces of 1-point functions associated to intertwining operators for Virasoro minimal model VOAs. We examine all representations of dimension less than four associated to irreducible modules for minimal models, and determine when the kernel of these representations is a congruence or noncongruence subgroup of the modular group. Arithmetic criteria are given on the indexing of the irreducible modules for minimal models that imply the associated modular group representation has a noncongruence kernel, independent of the dimension of the representation. The algebraic structure of the spaces of 1-point functions for intertwining operators is also studied, via a comparison with the associated spaces of holomorphic vector-valued modular forms. *
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