The project focuses on the conjectural relations between certain arithmetic invariants attached to Galois representations, in particular the cohomological invariants called Selmer groups, and the order of vanishing at integers of either the L-functions, or the p-adic L-functions attached to those Galois representations. The main objective is to prove that the rank of the Selmer group of a Galois representation attached to an automorphic form for a unitary group is at least equal to the order of vanishing at 0 of the corresponding p-adic L-function (one actually expects that the equality holds). The strategy consists in a study of the geometry of the eigenvariety, which is the universal family of automorphic forms, at a point attached to the given Galois representation, and in the construction of a family of p-adic L-functions on that eigenvariety. The project has an other aspect, which consists in reformulating and generalizing the conjectures relating p-adic L-functions and Galois cohomology.
While the automorphic methods used in this project are very promising, it is not expected that they alone will solve the vast array of conjectures and questions concerning the relations between p-adic L-functions and Galois cohomology. Many tools and ideas from various parts of mathematics will be needed to do so. One of the aims of the PI in reformulating those conjectures is to separate more clearly what part can be done with which methods or combination of methods, and to foster a greater involvement of mathematicians in other areas (e.g., theory of transcendence) in the work on those conjectures.
In the last year of this project, the PI has completed most of the goals of the initial proposal concerning p-adic families of automorphic forms and L-functions, and explored new avenues of research concerning the theory of modular forms modulo p. Concerning the first aspect., the PI has completed and released a book, which he intends to submit for pubcation soon. This book has two aims. One is to present in a single place a comprehensive and complete treatment of the modern theory of families of p-adic modular forms and of their p-adic L-functions. While the progresses have been quick and manifold in this discipline, they have been disseminated in journal articles, or often only in unpublished preprints without a wide diffusion. There has been no book with a similar scope since the books of Gouvêa and Hida fifteen years ago, which represent an earlier stage of the theory, and I think the subject was in need for a new syntehsis book. Thus the book contains several chapters worth of material which is not new, except perhaps for the presentation and the care in giving complete argumens in large generalities. The author hopes that this presentation will be useful for students entering in the subject (and the circulation of some pre-released versions indicate it will), but also will provide a solid ground for specialists on which to base future work. This leads me to the second aim of this book, which is to present original research of the PI, especially on adjoint of p-adic L-functions. A previous tenttaive to write this material before the book ran against the lack of satisfying foundations in many places. This book, and 6 research articles written by the Pi during the grant, have improved the understanding of p-adic L-functions and eigenvarieties, and paves the way for further work on the Birch and Swinnerton-Dyer conjecture. The second aspect of the Pi's work has concerned another theory, the theory of modular forms modulo p. The PI has worked on various aspect on this theory, often in collaboration. An important result, obtained with C. Khare, has been the determination of the structure of the Hecke algebras of modular forms modulo p, a problem that was open since the 1970's.