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Galois Representations, Monodromy Groups, and Motives

伽罗瓦表示、单调群和动机

基本信息

批准号：
1700759
负责人：
Stefan Patrikis
金额：
$ 14万
依托单位：
University of Utah
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2020-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700759&HistoricalAwards=false
关键词：
Galois Representations Monodromy Groups Motives

项目摘要

Throughout the mathematical sciences, symmetry is a powerful organizing principle. In number theory, which studies the integers, especially the prime numbers, there are no manifest symmetries of the kind encountered in more intuitive, geometric settings (symmetry of a square under 90-degree rotations, etc.). Nevertheless, much of our deepest knowledge of the primes comes from their connections with the "symmetries" of polynomial equations. This project focuses on the study of Galois representations, which are natural packages for this information about the prime numbers. One of the central programs of modern number theory, initiated by Robert Langlands, proposes new ways to "unpack" Galois representations by relating them to remarkably different mathematical objects arising in geometry. Although they are essentially about the prime numbers, these relations have even had reverberations in fundamental physics. This project involves both constructing new Galois representations and unpacking the information they contain in the light of Langlands' conjectures.This project comprises two research programs. The first will study deformations of Galois representations valued in arbitrary reductive groups, with a view toward establishing in as great generality as possible the existence of geometric p-adic deformations of a given mod p representation. The work is likely to furnish steps toward generalizations of Serre's modularity conjecture to general reductive groups. Such geometric p-adic representations are expected to be the (semi-simple) Galois representations occurring in the cohomology of algebraic varieties over number fields. The second program aims to study a number of questions of Fontaine-Mazur type, seeking to establish the motivic origin of certain Galois representations or, in a more geometric setting, local systems over a curve. The work will require elaborating the motivic structures underlying basic objects of the geometric Langlands program, a problem that should be of long-term independent interest even for the arithmetic Langlands program.

在整个数学科学中，对称性是一个强大的组织原则。在研究整数（尤其是素数）的数论中，不存在在更直观的几何设置中遇到的那种明显的对称性（正方形在 90 度旋转下的对称性等）。尽管如此，我们对素数的最深入的了解大多来自于它们与多项式方程“对称性”的联系。该项目重点研究伽罗瓦表示，它是有关素数信息的自然封装。由罗伯特·朗兰兹发起的现代数论的核心项目之一，提出了通过将伽罗瓦表示与几何中出现的截然不同的数学对象联系起来来“解开”伽罗瓦表示的新方法。尽管它们本质上是关于素数的，但这些关系甚至在基础物理学中产生了影响。该项目涉及构建新的伽罗瓦表示并根据朗兰兹猜想解压缩其中包含的信息。该项目包括两个研究项目。第一个将研究在任意还原群中评估的伽罗瓦表示的变形，以尽可能普遍地建立给定 mod p 表示的几何 p 进变形的存在。这项工作可能会为将塞尔的模块化猜想推广到一般还原群提供步骤。这种几何 p-adic 表示预计将成为数域上代数簇的上同调中出现的（半简单）伽罗瓦表示。第二个项目旨在研究一些 Fontaine-Mazur 类型的问题，寻求建立某些伽罗瓦表示的动机起源，或者在更几何的环境中，建立曲线上的局部系统。这项工作需要详细阐述几何朗兰兹纲领的基本对象背后的动机结构，即使对于算术朗兰兹纲领来说，这个问题也应该具有长期的独立兴趣。