Modular forms for noncongruence subgroups: an overview

Winnie Li , Penn State University
Fine Hall 314

The two most important tools used to study the arithmetic of modular forms for congruence subgroups are the Hecke theory and l-adic Galois representations. Unlike their congruence counterpart, the arithmetic for noncongruence modular forms remains mysterious. A main reason is the lack of efficient Hecke operators. Based on their numerical evidence, Atkin and Swinnerton-Dyer proposed a substitute for the Hecke eigenform at good primes, expressed in terms of 3-term congruence relations. Later Scholl attached l-adic Galois representations to the space of noncongruence forms. These representations are motivic, and hence should correspond to automorphic representations according to the Langlands philosophy. The automorphy of Scholl representations is established only for very special cases. The area of noncongruence modular forms is a fertile ground to explore. In this talk we shall review the development of the field and its current status.