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

在过去的几年里，数论取得了许多重大进展。算术几何、模形式理论和伽罗瓦表示的结果已经给出了费马最后定理的证明和向$p$-ady Birch-Swinnerton-Dyer猜想(BSD)的基本进展，仅举两例。本项目提出的研究旨在继续这一进展。PI建议研究自同构形式、伽罗瓦表示和其$L$-函数的值之间的许多方面的联系，特别目的是在BSD、Bloch-Kato猜想和岩泽自同构伽罗瓦表示理论方面取得进展，以及回答与自同构形式相关的伽罗瓦表示的基本问题。他们的项目集中在自同构形式和伽罗瓦表示理论中的$p$-ady方法。通过结合他们不同的专业知识，他们建议考虑以下标题下的一些具体问题：$p$-adic Eisenstein测度及其专门化，岩泽的$\u$-不变量，$L$-值的非零模$p$，酉群的Eisenstein理想，$p$-adic自同构形式的几何构造，$p$-adic欧拉系的构造，内窥镜同余，Galois表示和Shimura簇。这个项目将增强我们对自同构形式、伽罗瓦表示及其$L$-函数之间的深层联系的了解--这是数论的一个中心焦点--并对我们对一般数学的理解有重要的影响。两次讲习班、一次期末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 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.