Noncongruence Modular Farms and Supercongruences

Louisiana State University, Baton Rouge LA

Investigators

Abstract

Modular forms are spectacular functions that are highly symmetric. They occur in abundance throughout mathematics and physics. For more than one century, the theory of modular forms has being playing a central role in number theory, as witnessed in the proof of Fermat's Last Theorem. The symmetries of modular forms are captured by the elements of the special linear group of degree 2 over the ring of integers. Among all modular forms, majority of them are noncongruence in the sense that their symmetries cannot be described by congruences. The study of noncongruence modular forms has been fallen behind its congruence counterpart due to the lack of a satisfactory Hecke theory. However, recent progresses on noncongruence modular forms have revealed their rich connections with several fruitful research frontiers: p-adic modular forms, automorphic forms, and Galois representations. The theory of noncongruence modular forms has far reaching impacts on other areas like the conformal field theory and combinatorics. The project aims at a further theoretic development of noncongruence modular forms with applications to problems like supercongruences, which are special congruences satisfied by many interesting combinatorial or arithmetic sequences. The scientific objectives are: study when Galois representations attached to noncongruence modular forms are related to automorphic forms via Langlands correspondence; understand a fundamental conjecture that characterizes genuine noncongruence modular forms; explore supercongruences using the perspective of Atkin and Swinnerton-Dyer congruences that were originated in the study of noncongruence modular forms. Currently, the proofs of supercongruences often require the finding of auxiliary identities, which procedure often involves intriguing guesses. A more conceptual understanding of supercongruences may shed lights on how to search for these identities systematically. The research outcomes will be published by high quality journals and disseminated at various conferences and seminars. The PI will continue to widen the impacts of the her research program by mentoring undergraduate students, graduate students, and postdocs, as well as organizing scientific conferences. In recent years, the PI has been seriously involved in activities to promote the advancement of women in mathematics. She was a group co-leader for Banff International Research Station workshops on Women in Numbers (WIN) in 2008 and WIN2 in 2011 and is one of the organizers for a coming WIN 3 conference in 2014.