On the Fourier coefficients of 2-dimensional vector-valued modular forms

Geoffrey Mason

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arXiv · 1009.0781

On the Fourier coefficients of 2-dimensional vector-valued modular forms

Geoffrey Mason

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Abstract

Let $ρ: SL(2,\mathbb{Z})\to GL(2,\mathbb{C})$ be an irreducible representation of the modular group such that $ρ(T)$ has finite order $N$. We study holomorphic vector-valued modular forms $F(τ)$ of integral weight associated to $ρ$ which have \emph{rational} Fourier coefficients. (These span the complex space of all integral weight vector-valued modular forms associated to $ρ$.) As a special case of the main Theorem, we prove that if $N$ does \emph{not} divide 120 then every nonzero $F(τ)$ has Fourier coefficients with \emph{unbounded denominators}.

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Geoffrey Mason. 2010-09-03. On the Fourier coefficients of 2-dimensional vector-valued modular forms. https://arxiv.org/abs/1009.0781

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