Number theory has seen many significant advances in the past few years. Results from arithmetic geometry and the theories of modular forms and Galois representations have yielded a proof Fermat's Last Theorem and fundamental advances towards the $p$-adic Birch-Swinnerton-Dyer Conjecture (BSD), to name two. The research proposed in this project aims to continue this progress. The PI's propose to investigate many aspects of the connections between automorphic forms, Galois representations, and values of their $L$-functions, with the particular aim of making advances towards BSD, Bloch-Kato conjectures, and the Iwasawa Theory of automorphic Galois representations, as well as answering fundamental questions about the Galois representations associated to automorphic forms. Their project focuses on $p$-adic methods in the theory of automorphic forms and Galois representations. By combining their various expertise, they propose to consider a number of specific problems that fall under the following headings: $p$-adic Eisenstein measures and their specializations, Iwasawa's $mu$-invariants, Non-vanishing modulo $p$ of $L$-values, Eisenstein ideals for unitary groups, Geometric construction of $p$-adic automorphic forms, $p$-adic construction of Euler systems, Endoscopic congruences, Galois representations and Shimura varieties.
This project will enhance our knowledge of the deep links between automorphic forms, Galois representations, and their $L$-functions - a central focus of number theory - as well as have significant consequences for our understanding of mathematics in general. Two workshops, a final conference, and graduate and post-doctoral advising will have an important impact on the formation of new researchers in the field and on the promotion of new collaborations.