FRG: Collaborative Research: Automorphic Forms, Galois Representations, and Special Values of L-functions.

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Number theory has seen many significant advances in the past few years.Results from arithmetic geometry and the theories of modular forms andGalois representations have yielded a proof Fermat's Last Theorem andfundamental 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 theconnections 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 theGalois representations associated to automorphic forms. Their projectfocuses on $p$-adic methods in the theory of automorphic forms and Galois representations. By combining their various expertise, they propose toconsider a number of specific problems that fall under the followingheadings: $p$-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 $p$-adicautomorphic forms, $p$-adic construction of Euler systems, Endoscopiccongruences, Galois representations and Shimura varieties.This project will enhance our knowledge of the deep links betweenautomorphic forms, Galois representations, and their $L$-functions - acentral focus of number theory - as well as have significant consequences for our understanding of mathematics in general. Two workshops, a finalconference, and graduate and post-doctoral advising will have animportant impact on the formation of new researchers in the field and onthe promotion of new collaborations.