The unbounded denominator conjecture for the noncongruence subgroups of index $7$

TL;DR

This paper proves the unbounded denominator conjecture for minimal index noncongruence subgroups of the modular group and analyzes Fourier coefficients of Eisenstein series for these groups.

Contribution

It provides the first proof of the unbounded denominator conjecture for specific noncongruence subgroups and studies their Eisenstein series Fourier coefficients.

Findings

Proof of the unbounded denominator conjecture for minimal index noncongruence subgroups

Analysis of Fourier coefficients of Eisenstein series for these groups

Insights into the structure of modular forms for noncongruence subgroups

Abstract

We study modular forms for the minimal index noncongruence subgroups of the modular group. Our main theorem is a proof of the unbounded denominator conjecture for these groups, and we also provide a study of the Fourier coefficients of Eisenstein series for one of these minimal groups.