The domain of research of this project is the arithmetic of p-adic automorphic forms, their Galois representations and L-functions. Classical automorphic forms for a group G are functions on the adelic points of a reductive group G that satisfy nice transformation properties. Their arithmetic theory has yielded significant advances in the past few years:a proof Fermat's Last Theorem and a proof the Sato-Tate conjecture, to name two. The p-adic notions alluded above is a build-in concept to study congruences between these classical objects. Urban's proposal is a continuation on his work related to the construction and the study of congruences between Eisenstein series and cuspidal automorphic forms of various weights and levels and their links with p-adic L-functions and certain arithmetically defined groups called Selmer groups. Urban proposes to 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. Here is a list of several topics that this project will deal with: (1) Dimension of irreducible components of Eigenvarieties (2) Galois representations for GL(n) and torsion classes for the cohomology of U(n,n), (3) p-adic deformations of holomorphic and nearly holomorphic automorphic forms, (4) Construction of p-adic measures attached to L-functions and Eisenstein series, (5) p-adic Euler system and p-adic L-functions.
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. Urban intends to write a book on the general theory of p-adic automorphic representations to report on the many developments that has known the past decade. Half of the book will be devoted to the theory for general reductive group. In the second half, he will describe some applications with examples and propose directions of future research. In particular, important conjectures of the theory will be described in detail. This book will be intended mainly to graduate student and researchers. Urban works jointly with some PhD students and recent post-doctoral mathematicians on some of his projects by organizing seminar and workshops. He also wants to enhance the training of these young mathematicians and facilitate interactions with highly qualified mathematicians he will invite on a regular basis. Such an activity, his book project and the participation by Urban and his students in other seminars throughout the country and in international conferences will be very useful for the dissemination of the advances resulting from this project.
This research project was about studying arithmetical properties of modular forms, their Galois representations and the consequences to study number theory questions using p-adic methods. What we call p-adic methods belongs to a general framework to study divisibilities of natural numbers by high powers of a prime number p. Developing these theories, as done in this project, has for target important arithmetic applications which are met in some cases. As a result, important progress towards a famous conjecture of Birch and Swinnerton-Dyer on the arithmetic of elliptic curves have been made. However, there are more general application to what is called class number formulas which are fundamentals to explain number theory phenomena. Understanding these deep arithmetic questions could have very important applications to cryptography and to understand prime numbers in general. The p-adic methods which have been developed in this project have other arithmetic applications which are used by other mathematicians in their research. Other than obtaining new knowledge in the arithmetic theory of automorphic forms, the PI has given several lectures in international conferences to report on his work. He has also trained new PhD students and postdocs on works related to this project. The PI has also started to write a book on these p-adic techniques to disseminate them for a more general audience.
