Arithmetic Questions in the Theory of Linear Algebraic Groups

线性代数群理论中的算术问题

• 批准号: 2154408
• 负责人: Igor Rapinchuk
• 金额: $ 24.13万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154408&HistoricalAwards=false
• 关键词: Arithmetic; Questions; Theory; Linear; Algebraic

Linear algebraic groups are groups of matrices that are described by polynomial equations. Such groups arise as groups of symmetries of various objects and are ubiquitous across many areas of mathematics, including algebraic geometry, number theory, and mathematical physics. In the arithmetic context, work done over the last six decades has resulted in a well-developed theory of linear algebraic groups over the rational numbers and other similar fields. While activity in this area is still ongoing, over the last ten years various problems in Lie group theory, arithmetic geometry, and other subjects have led to significant interest in the properties of algebraic groups over fields of geometric origin. Building on previous work, the research program will investigate the arithmetic, geometric, and structural aspects of algebraic groups over such higher-dimensional fields, with a particular focus on various finiteness properties. Mentoring graduate students and developing courses at the undergraduate and graduate levels will be an integral part of this work. In addition, a book project will be undertaken to open up recent developments in the emerging arithmetic theory of algebraic groups over higher-dimensional fields to a broader audience. The project is a multi-faceted research program in the study of algebraic groups over higher-dimensional fields. The work will focus on the following three directions: the analysis of algebraic groups with good reduction and applications to local-global principles, the study of finiteness properties of unramified cohomology, and the investigation of rigidity phenomena for abstract homomorphisms of algebraic groups. A major goal in the study of groups with good reduction will be to make progress on a finiteness conjecture for forms of reductive algebraic groups over finitely generated fields having good reduction with respect to divisorial sets of discrete valuations. This work will significantly expand the scope of previous results, which dealt mainly with groups over fraction fields of Dedekind rings, and will also have important consequences for the properness of the global-to-local map in the Galois cohomology of algebraic groups. It turns out that, for certain types of groups, this finiteness conjecture is closely related to finiteness properties of unramified cohomology. As a result, one of the objectives will be to establish the expected finiteness of unramified cohomology in degree three for surfaces and certain higher-dimensional varieties over global fields. Concerning abstract homomorphisms, the goal will be to develop a substantial generalization of methods introduced in previous work to resolve a longstanding conjecture of Borel and Tits for all absolutely almost simple groups over infinite fields of relative rank at least two.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

线性代数群是由多项式方程描述的矩阵组。这些群作为各种对象的对称群而出现，并且在许多数学领域中普遍存在，包括代数几何、数论和数学物理。在算术背景下，过去六十年所做的工作已经产生了有理数和其他类似领域的线性代数群的完善理论。虽然这一领域的活动仍在继续，但在过去十年中，李群理论、算术几何和其他学科中的各种问题引起了人们对几何起源域上代数群的性质的极大兴趣。在之前工作的基础上，该研究计划将研究此类高维域上代数群的算术、几何和结构方面，特别关注各种有限性性质。指导研究生以及开发本科生和研究生课程将是这项工作的一个组成部分。此外，还将开展一个图书项目，向更广泛的读者开放高维领域中新兴代数群算术理论的最新发展。该项目是一个多方面的研究项目，旨在研究高维领域的代数群。工作将集中在以下三个方向：具有良好约简性的代数群分析以及局部全局原理的应用、无枝上同调的有限性研究以及代数群抽象同态的刚性现象研究。研究具有良好约简的群的一个主要目标是在有限生成域上的约简代数群形式的有限性猜想上取得进展，该有限生成域对于离散估值的除数集具有良好的约简。这项工作将显着扩展先前结果的范围，这些结果主要处理戴德金环分数域上的群，并且还将对代数群伽罗瓦上同调中全局到局部映射的正确性产生重要影响。事实证明，对于某些类型的群，这种有限性猜想与无枝上同调的有限性性质密切相关。因此，目标之一是在全局域上建立曲面和某些高维簇的三阶无分支上同调的预期有限性。关于抽象同态，目标是对先前工作中引入的方法进行实质性概括，以解决长期存在的 Borel 和 Tits 猜想，适用于相对等级至少为 2 的无限域上的所有绝对几乎简单的群。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。