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