Modular Forms for Noncongruence Subgroups

Iowa State University, Ames IA

Investigators

Abstract

In recent years, stunning advances have been made in the study of modular forms for congruence subgroups: the settlements of the Taniyama-Shimura-Weil conjecture and Serre's conjecture, to name a few. On modular forms for noncongruence subgroups, rapid progress has also been made recently. While our knowledge of noncongruence modular forms is still in its relative infancy, the subject has already shown its rich connections with several fruitful research frontiers: classical modular forms, Galois representations, the Langlands program, and p-adic modular forms. The general aim of this proposal is to continue the development of noncongruence modular forms by the PI and her collaborators. The proposal contains 3 objectives: 1) To study when Galois representations attached to noncongruence cusp forms constructed by A. Scholl are related to classical automorphic forms via Langlands correspondence, as well as the applications of a relation. 2) To understand a fundamental conjecture which asserts that Fourier coefficients of genuine noncongruence modular forms have unbounded denominators if all coefficients are algebraic. 3) To explore p-adic properties of noncongruence modular forms and their applications to other research areas. Starting from the time of the Greeks, many great problems in number theory have challenged intellectual minds, and their considerations have provided numerous useful applications in turn. This proposal emphasizes theoretic developments, broad applications, as well as educating students. As a developing area, the theory of noncongruence modular form contains a wide spectrum of topics. Some suitable projects will be incorporated in the learning and training of graduate and advanced undergraduate students. The outcome will be disseminated widely through referred journal articles, seminar and conference talks, as well as topics for graduate courses.