Geometry of moduli stacks of Galois representations

伽罗瓦表示的模栈的几何

• 批准号: 2302623
• 负责人: Daniel Le
• 金额: $ 18.5万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302623&HistoricalAwards=false
• 关键词: Geometry; moduli; stacks; Galois; representations

A central problem in number theory concerns solving polynomial equations in the whole numbers and closely related number systems. Solving these so-called Diophantine equations is typically very difficult, and this difficulty is a source of security for encryption algorithms that, for instance, guarantee private communication on the internet. Moreover, the richness of these Diophantine problems has profound connections to other branches of mathematics. One can fruitfully study the set of solutions to these equations through its geometry and dynamics, repackaging them as a Galois representation. In the 1970s, Robert Langlands made far-reaching predictions that connect Galois representations to mathematical physics and harmonic analysis. A p-adic version of Langlands' conjectures, which incorporates divisibility by powers of a prime, has led to exciting progress on Langlands' original conjectures and, consequently, a range of classical number-theoretic questions like Fermat's Last Theorem. Further analogues of Langlands' conjectures in other contexts have spurred activity in representation theory, algebraic geometry, and modern physics. More recently, a number of mathematicians have proposed a unifying categorical framework for these different versions of Langlands' conjectures. This project aims to study the p-adic version of Langlands' conjectures through the categorical framework by modelling the space of Galois representations. The geometry of the space of Galois representations is a new and fertile research direction that provides opportunities to train younger researchers and to connect communities in number theory and geometric representation theory. The PI also plans to recruit and train researchers through outreach, by disseminating accessible background material on number theory aimed at beginning graduate students, and by organizing local and regional seminars and conferences.The goal of this project is to investigate the hypothetical mod p and p-adic Langlands correspondences that relate Galois representations and modular and integral representations of p-adic reductive groups. This correspondence is conjecturally mediated by sheaves on stacks of Galois representations. The Principal Investigator will construct local models for stacks of Galois representations to analyze these conjectural sheaves and answer questions around modularity lifting, the Breuil-Mezard conjecture, the weight part of Serre's conjecture, and the mod p cohomology of locally symmetric spaces using the Taylor-Wiles method. These investigations will allow for applications to and from modular and geometric representation theory. This project gives undergraduate and graduate students the opportunity to probe local models via combinatorial methods and computer algebra packages.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.

数字理论中的一个中心问题涉及在整体数字和密切相关的数字系统中求解多项式方程。求解这些所谓的借用方程通常非常困难，而且这种困难是加密算法的安全性，例如，保证在互联网上保证私人通信。此外，这些养生问题的丰富性与数学的其他分支有着密切的联系。人们可以通过其几何形状和动力学来富有成果研究这些方程式的解决方案，从而将它们重新包装为Galois表示。在1970年代，罗伯特·兰兰兹（Robert Langlands）做出了深远的预测，将galois表示与数学物理学和谐波分析联系起来。 Langlands猜想的P-ADIC版本结合了质量的能力，这使Langlands的原始猜想取得了令人兴奋的进步，因此，诸如Fermat的Last Therorem之类的一系列经典数字理论问题。在其他情况下，兰兰兹的猜想的进一步类似于表示理论，代数几何和现代物理学的活动。最近，许多数学家提出了这些不同版本的兰兰兹猜想的统一分类框架。该项目旨在通过对Galois表示的空间进行模拟，通过分类框架来研究Langlands的P-Adic版本。加洛伊斯表示空间的几何形状是一个新的肥沃的研究方向，它为培训年轻研究人员并连接数字理论和几何表示理论的社区提供了机会。 PI还计划通过宣传来招募和培训研究人员，通过传播旨在初学者的数字理论的可访问背景材料，并通过组织本地和地区的研讨会和会议。该项目的目的是调查假设的Mod P和P-Adic Langlands对应关系，与PALOIS代表和模块化组成的p-aducticals conductive conductive conductive conductive conductive conductive codductive codductive reductive codductive codductive。这种对应关系是由盖岛表示堆栈上的带轮介导的。首席研究者将构建用于GALOIS表示堆栈的本地模型，以分析这些猜想的滑轮，并通过使用泰勒 - Wiles-Wiles Method方法来回答有关模块化，Breuil-Mezard的猜想，Serre猜想的重量部分以及局部对称空间的MOD P共同体的问题。这些研究将允许使用模块化和几何表示理论的应用。该项目为本科生和研究生提供了通过组合方法和计算机代数软件包来探究本地模型的机会。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估的评估值得支持的。