The subject of this proposal is research in the area of congruences, or p-adic analysis, related to modular forms. The research area of this proposal combines and provides links between highly abstract and fundamental objects such as the Eigencurve recently discovered by Mazur and Coleman, which describes all the two-dimensional Galois representations in p-adic families, and various elementary number theoretic objects such as the partition function recently investigated using the methods of the p-adic theory of modular forms by Ono and Ahlgren. Some open problems in this area are fairly new. In particular, in a recently published book "The Web of Modularity: Arithmetic of the Coefficients of Modular Forms and q-Series" Ono provides many new insights and results along with a variety of open problems. There are also long-standing conjectures, for example, those formulated by Atkin almost four decades ago pertaining to certain congruences for the Fourier coefficients of the modular invariant. Sometimes the p-adic setting sheds a new light on questions which come from the complex-analytic theory of modular forms (e.g., a question about connections between Borcherds and Shimura maps). This proposal consists of several distinct projects that address problems of these kinds. These projects are at present in different stages of development. In all cases the principal investigator has already obtained certain results, and there are reasons to expect further progress.
Partly motivated by congruences found by Ramanujan and pioneered in works of Atkin, Serre, and Swinnerton-Dyer, this area of research has steadily produced significant results during the last forty years. Many important achievements in theoretical and applied number theory in the last few decades, including the celebrated proof of Fermat's Last Theorem and elliptic curve cryptography, are closely related to this area of research. Results from this kind of number theory find their applications in various areas of mathematics such as elliptic genera, arithmetic geometry, finite fields, as well as certain areas of contemporary physics such as mirror symmetry, conformal invariance and string theory. Sometimes the applications and connections are discovered years after a result is obtained. Therefore, although no results of this scope or specific connections to other fields are claimed in this proposal, contributions to this area of research may well have future applications.
