Fourier coefficients of noncongruence cuspforms

TL;DR

This paper establishes a criterion linking bounded denominators of Fourier coefficients of certain cusp forms to their congruence nature, under specific algebraic and geometric conditions.

Contribution

It proves that for cusp forms with rational Fourier coefficients, bounded denominators imply the form is a congruence modular form, under the given assumptions.

Findings

Bounded denominators characterize congruence forms.

The result applies to forms with one-dimensional cusp form spaces.

The proof relies on properties of modular curves over $\

Abstract

Given a finite index subgroup of $SL_2(\mathbb Z)$ with modular curve defined over $\mathbb Q$, under the assumption that the space of weight $k$ ($ \ge 2$) cusp forms is $1$-dimensional, we show that a form in this space with Fourier coefficients in $\mathbb Q$ has bounded denominators if and only if it is a congruence modular form.