Finite index subgroups of the modular group and their modular forms

TL;DR

This survey explores the properties and differences of congruence and noncongruence subgroups of the modular group, focusing on their modular forms and key phenomena like unbounded denominators and Galois representations.

Contribution

It provides a comprehensive overview of noncongruence subgroups and their modular forms, highlighting recent developments and open problems in the field.

Findings

Noncongruence subgroups are more prevalent among finite index subgroups.

Noncongruence modular forms exhibit unbounded denominators.

Galois representations from noncongruence cusp forms show modularity.

Abstract

Classically, congruence subgroups of the modular group, which can be described by congruence relations, play important roles in group theory and modular forms. In reality, the majority of finite index subgroups of the modular group are noncongruence. These groups as well as their modular forms are central players of this survey article. Differences between congruence and noncongruence subgroups and modular forms will be discussed. We will mainly focus on three interesting aspects of modular forms for noncongruence subgroups: the unbounded denominator property, modularity of the Galois representation arising from noncongruence cuspforms, and Atkin and Swinnerton-Dyer congruences.