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

数论在过去几年中取得了许多重大进展。等差几何、模形式理论和伽罗瓦表示的结果已经证明了费马大定理，并对$p$进阶Birch-Swinnerton-Dyer猜想（BSD）取得了根本性的进展，举两个例子。本项目提出的研究旨在继续这一进展。PI计划研究自同构形式、伽罗瓦表示及其$L$-函数的值之间的联系的许多方面，其具体目标是在BSD、Bloch-Kato猜想和Iwasawa自同构伽罗瓦表示理论方面取得进展，并回答与自同构形式相关的伽罗瓦表示的基本问题。他们的项目主要集中在自同构形式和伽罗瓦表示理论中的$p$-方法。通过结合他们的各种专业知识，他们提出考虑一些具体的问题，这些问题属于以下标题：$p$-一元爱森斯坦测度及其专门化，Iwasawa的$\mu$-不变量，$L$值的非消失模$p$，一元群的爱森斯坦理想，$p$-一元自同构形式的几何构造，欧拉系统的$p$-一元构造，内视镜同余，伽罗瓦表示和志村变体。该项目将增强我们对自同构形式、伽罗瓦表示及其L函数（数论的中心焦点）之间的深层联系的认识，并对我们对数学的总体理解产生重大影响。两次研讨会，一次最终会议，以及研究生和博士后指导将对该领域新研究人员的形成和促进新的合作产生重要影响。

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.