On modular forms for some noncongruence arithmetic subgroups

TL;DR

This paper investigates the properties of modular forms for noncongruence subgroups, focusing on the unbounded denominator property, and provides partial answers through the construction of specific character groups.

Contribution

It introduces special finite index subgroups called character groups and analyzes their modular forms, advancing understanding of the unbounded denominator property for noncongruence forms.

Findings

Constructed specific character groups of SL_2(Z)

Provided partial evidence that noncongruence forms can have unbounded denominators

Discussed properties distinguishing noncongruence modular forms from congruence ones

Abstract

In this paper, we consider modular forms for finite index subgroups of the modular group whose Fourier coefficients are algebraic. It is well-known that the Fourier coefficients of any holomorphic modular form for a congruence subgroup (with algebraic coefficients) have bounded denominators. It was observed by Atkin and Swinnerton-Dyer that this is no longer true for modular forms for noncongruence subgroups and they pointed out that unbounded denominator property is a clear distinction between modular forms for noncongruence and congruence modular forms. It is an open question whether genuine noncongruence modular forms (with algebraic coefficients) always satisfy the unbounded denominator property. Here, we give a partial positive answer to the above open question by constructing special finite index subgroups of SL_2(Z) called character groups and discuss the properties of modular forms for some groups of this kind.