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

FRG：协作研究：自守形式、伽罗瓦表示和 L 函数的特殊值。

• 批准号: 0456298
• 负责人: Eric Jean-Paul Urban
• 金额: --
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-01 至 2011-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0456298&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Automorphic; Forms

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.

数论在过去的几年里取得了许多重大的进展。算术几何、模形式和伽罗瓦表示理论的结果证明了费马大定理，并对p$-adic Birch-Swinnerton-Dyer猜想（BSD）取得了根本性的进展。本项目中提出的研究旨在继续这一进展。PI的建议，以调查的许多方面的theconnections之间的自守形式，伽罗瓦表示，和价值观的$L$-功能，与特定的目的，使进步的BSD，布洛赫-加藤结构，和岩泽理论的自守伽罗瓦表示，以及回答有关的基本问题伽罗瓦表示与自守形式。他们的项目侧重于自守形式和伽罗瓦表示理论中的$p$-adic方法。通过结合他们的各种专门知识，他们建议考虑一些具体问题，这些问题属于以下几个方面：p$-adic Eisenstein测度及其特殊化，Iwasawa的$\mu$-不变量，L$-值的模p$非零，酉群的Eisenstein理想，p$-adic自守形式的几何构造，Euler系统的p$-adic构造，Endoscopic同余，伽罗瓦表示和志村品种。这个项目将提高我们的知识之间的深链接自守形式，伽罗瓦表示，和他们的$L$-功能-一个中心的重点数论-以及有重大影响，我们的理解数学一般。两场研讨会、一场最终会议以及研究生和博士后咨询将对该领域新研究人员的形成和新合作的促进产生重要影响。