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

Mason, Geoffrey

doi:10.1090/S0002-9939-2011-11098-0

On the Fourier coefficients of 2-dimensional vector-valued modular forms
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Proc. Amer. Math. Soc. 140 (2012), 1921-1930 Request permission

Abstract:

Let $\rho : SL(2, \mathbb {Z}) \rightarrow GL(2, \mathbb {C})$ 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 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 not divide $120$, then every nonzero $F(\tau )$ has Fourier coefficients with unbounded denominators.

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