Fourier coefficients of vector-valued modular forms of dimension 2

TL;DR

The paper proves that 2-dimensional vector-valued modular forms with rational Fourier coefficients and bounded denominators are classical modular forms on a congruence subgroup, establishing a significant structural result.

Contribution

It demonstrates that such vector-valued modular forms are necessarily classical modular forms, linking rationality and bounded denominators to classical modularity.

Findings

Vector-valued modular forms with rational Fourier coefficients are classical forms.

Bounded denominators imply classical modularity.

Results apply to forms on $SL_2(Z)$ with rational Fourier coefficients.

Abstract

We prove the following theorem. Suppose that $F=(f_1, f_2)$ is a 2-dimensional vector-valued modular form on $SL_2(Z)$ 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.