Noncongruence modular forms and modularity
by
Wen-Ching Winnie Li
Pennsylvania State University

Unlike classical modular forms, the arithmetic of noncongruence modular forms is not well-understood. In their pioneering work, Atkin and Swinnerton-Dyer suggested very interesting congruence relations on the Fourier coefficients of noncongruence forms. Scholl associated l-adic Galois representations to such forms. In this survey talk we shall review the recent progress on the arithmetic properties of noncongruence forms, including congruence relations between the Fourier coefficients of noncongruence and congruence forms, the modularity of the Scholl representations, and the unbounded denominator criterion for noncongruence forms.

Date received: April 8, 2008