This research project in number theory studies arithmetic properties of automorphic forms, which are functions that satisfy certain transformation properties. More specifically, the project is to study divisibility properties of these mathematical objects by high powers of a given prime number. The study of these properties has yielded significant advances in number theory in the past few years: a proof Fermat's Last Theorem, a proof the Sato-Tate conjecture, and a proof of the Iwasawa main conjecture for elliptic curves. The work resulting from this project will give more insight into some of the current most important problems in number theory. This project will enhance our knowledge of the deep relationships between p-adic automorphic forms, Galois representations, and their p-adic L functions -- a central focus of number theory -- as well as have significant consequences for our understanding of mathematics in general. The project will involve the training of graduate students.
The domain of research of this project is the arithmetic of p-adic automorphic forms, their Galois representations, and L-functions. This project is a continuation of the PI's work related to the construction and the study of congruences between automorphic forms of various weights and levels and their links with p-adic L-functions and Selmer groups. The project will continue to build some of the foundations of the general theory of the p-adic automorphic forms and p-adic Eisenstein series with an eye one the important applications that will result. In particular, this theory applied to the case of unitary and symplectic groups will have important applications to the so-called p-adic Bloch-Kato conjecture and Birch and Swinnerton-Dyer conjecture. Studying the dimension of irreducible components of eigenvarieties, p-adic deformations of nearly over-convergent automorphic forms, critical p-adic L-functions, p-adic interpolation of automorphic periods, construction of p-adic measures attached to L-functions and Eisenstein series, p-adic Euler systems, and Kolyvagin systems and their link with p-adic L-functions are some of the topics that will be dealt with in this project.
