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.
The project was centered around the arithmetic properties of modular forms, which form an important class of functions that play a vital role in number theory, geometry, Combinatorics as manifested by the Grothendieck's profound program of Dessin d'enfant (child's drawing). In the last two and half decades, the world has also seen explosive interactions between the theory of modular forms and theoretical physics (in particular, String theory). There is a special category of modular forms called congruence modular forms, which are well-understood and have many fruitful applications. However, the majority of modular forms are not in this category and are said to be noncongruence. Noncongruence modular forms provide fertile testing grounds for many recent developments in Galois representation, Langlands' program, analytic tools in positive characteristic. One of the major outcome is a surprising connection between certain noncongruence modular forms and explicit `congruence' objects (known as automorphic forms) through the machinery of L-functions in number theory. This connection provides an important and convincing evidence to a long-standing conjecture which characterizes genuine noncongruence modular forms in terms of its arithmetic properties. This is another important outcome of the project. Besides, the PI also used various techniques developed for understanding noncongruence modular forms to tackle successfully open questions in other areas including supercongruences, hypergeometric series, abelian varieties related to triangle groups. The course of the investigation involved many aspects of training and mentoring. There was a Research Experience for Undergraduates (REU) component. The PI also mentored several graduate students, and postdoctorals through research projects. Two of the projects were carried out in women in number theory (WIN) workshops (2011 and 2014). The WIN program aims to promote the participantion of women in number theory which has been traditionally prodominated by male mathematicians. The PI disseminated the research outcomes by giving presentations at various professional occasions and organizing conferences including the 2014 WIN conference.
