Logarithmic vector-valued modular forms

2009/10/20 by Marvin I. Knopp, Knopp, Marvin, Geoffrey Mason +1 · 1 citation
Mathematics · #11F11 #11F99 #30F35 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0910.3976

openalex publication_date 2009/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider logarithmic vector- and matrix-valued modular forms of integral weight k associated with a p-dimensional representation ρ: SL₂(ℤ) → GL_p(ℂ) of the modular group, subject only to the condition that ρ(T) has eigenvalues of absolute value 1. The main result is the construction of meromorphic matrix-valued Poincaré series associated to ρ for all large enough weights. The component functions are logarithmic q-series, i.e., finite sums of products of q-series and powers of log q. We derive several consequences, in particular we show that the space H(ρ)=⊕_k H(k, ρ) of all holomorphic logarithmic vector-valued modular forms associated to ρ is a free module of rank p over the ring of classical holomorphic modular forms on SL₂(ℤ).

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