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# Fourier Coefficients of Vector-valued Modular Forms of Dimension $2$

Published:2014-04-05
Printed: Sep 2014
• Cameron Franc,
Department of Mathematics, University of California, Santa Cruz
• Geoffrey Mason,
Department of Mathematics, University of California, Santa Cruz
 Format: LaTeX MathJax PDF

## Abstract

We prove the following Theorem. Suppose that $F=(f_1, f_2)$ is a $2$-dimensional vector-valued modular form on $\operatorname{SL}_2(\mathbb{Z})$ whose component functions $f_1, f_2$ have rational Fourier coefficients with bounded denominators. Then $f_1$ and $f_2$ are classical modular forms on a congruence subgroup of the modular group.
 Keywords: vector-valued modular form, modular group, bounded denominators
 MSC Classifications: 11F41 - Automorphic forms on ${\rm GL}(2)$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 11G99 - None of the above, but in this section

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