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

TL;DR

This paper investigates the Fourier coefficients of holomorphic vector-valued modular forms associated with certain representations of the modular group, revealing conditions under which these coefficients have unbounded denominators.

Contribution

It proves that if the order of the representation's T-matrix does not divide 120, then all nonzero forms have Fourier coefficients with unbounded denominators.

Findings

Fourier coefficients can have bounded denominators only if the order divides 120.

The main theorem links the order of the representation to the boundedness of Fourier coefficient denominators.

Unbounded denominators occur for most representations outside specific divisibility conditions.

Abstract

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}.