The theory of vector-modular forms for the modular group

2013/10/16 by Terry Gannon, Gannon, Terry · 2 citations
Mathematics · #11F12 #34M50 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1310.4458

openalex publication_date 2013/10/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the modular group. Among other things,we construct vvmf for arbitrary multipliers, solve the Mittag-Leffler problem here, establish Serre duality and find a dimension formula for holomorphic vvmf, all in far greater generality than has been done elsewhere. More important, the new ideas involved are sufficiently simple and robust that this entire theory extends directly to any genus-0 Fuchsian group.

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