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

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

2010-09-03

arxiv.org/abs/1009.0781v1

Mathematics

Number Theory

Scientific paper

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

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