Departmental Colloquium
Winnie Li
Penn State University
Noncongruence modular forms and modularity
Abstract: Unlike congruence modular forms, the arithmetic of
noncongruence modular forms is not much understood, due to
the lack of Hecke operators. For noncongruence forms, Atkin and
Swinnerton-Dyer proposed a substitute of the familiar degree two recursive
relation satisfied by congruence Hecke eigenforms by three-term
congruence relations. In a certain situation, this yields very interesting
congruence relations between the Fourier coefficients of congruence and
noncongruence forms.
In this survey talk, we shall review the development of noncongruence
forms, discuss the progress on congruence relations, as well as the
unbounded denominator conjecture, which asserts that the algebraic
noncongruence forms are distinguished by its Fourier coefficients having
unbounded denominators. We shall see that the modularity of certain Galois
representations plays an essential role.
The colloquium is one of the talks of the Midwest Number Theory Days,
organized by Alina Carmen Cojocaru, Izzet Coskun,
Ramin Takloo-Bighash, and Jeremy Teitelbaum.
Please note the location: Lecture Center D-2.
Friday March 7, 2008 at 3:00 PM in Lecture Center D-2