2007/07/23 by Линг Лонг, Long, Ling
Mathematics · #11F11 #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.0707.3315
openalex publication_date 2007/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.