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群。这样的群作为各种对象的对称群出现，并且在数学的许多领域中无处不在，包括代数几何，数论和数学物理。在算术方面，过去60年的工作导致了有理数和其他类似领域的线性代数群的良好理论。虽然活动在这方面仍在进行中，在过去的十年中各种问题的李群理论，算术几何，和其他科目已导致显着的兴趣，性质的代数群领域的几何起源。在以前工作的基础上，该研究计划将研究代数群在这些高维域上的算术，几何和结构方面，特别关注各种有限性。指导研究生和在本科生和研究生一级开发课程将是这项工作的一个组成部分。此外，一本书的项目将进行开放的最新发展，在新兴的算术理论的代数群在高维领域，以更广泛的观众。该项目是一个多方面的研究计划，在高维领域的代数群的研究。工作将集中在以下三个方向：具有良好约化的代数群的分析及其在局部-整体原理中的应用，非分歧上同调的有限性性质的研究，以及代数群的抽象同态的刚性现象的研究。研究具有良好约化的群的一个主要目标是在有限生成域上的约化代数群形式的有限性猜想方面取得进展，该域相对于离散估值的整除集具有良好的约化。这项工作将显着扩大范围以前的结果，主要是处理组分数域的Dedekind环，也将有重要的后果，适当的全球到本地的映射在伽罗瓦上同调代数群。事实证明，对于某些类型的群，这个有限性猜想与非分歧上同调的有限性性质密切相关。其结果是，目标之一将是建立预期的有限性的非分歧上同调度为3的表面和某些高维品种在全球领域。关于抽象的同态，目标将是发展一个实质性的推广方法介绍了在以前的工作，以解决一个长期存在的猜想博雷尔和山雀为所有绝对几乎简单的群体在无限领域的相对秩至少两个。这个奖项反映了NSF的法定使命，并已被认为是值得支持通过评估使用该基金会的智力价值和更广泛的影响审查标准。