Noncongruence modular forms and unbounded denominators

Abstract: The modular group PSL2(Z) contains many noncongruence subgroups of finite index. In this talk we will explain some results on computing with the modular group, and in particular we will explain how to classify genus zero subgroups with a single cusp. Surprisingly, there are many such groups. Then we will discuss the unbounded denominator conjecture for some new cases of noncongruence subgroups of genus zero, using a method of Atkin and Swinnerton-Dyer, supplemented with some results on vector-valued modular forms. The most difficult step in this approach is to solve a system of diophantine equations defining an Artinian ideal. This is joint work with Andrew Fiori.

number theory

Audience: researchers in the topic

CRM-CICMA Québec Vermont Seminar Series

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