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.

数论中的一个中心问题是如何在整数和密切相关的数系中求解多项式方程。解决这些所谓的丢番图方程通常是非常困难的，而这种困难是加密算法的安全来源，例如，保证互联网上的私人通信。此外，丢番图问题的丰富性与数学的其他分支有着深刻的联系。我们可以通过几何学和动力学来研究这些方程的解，把它们重新包装成伽罗瓦表示法。在20世纪70年代，罗伯特·朗兰兹做出了影响深远的预测，将伽罗瓦表示法与数学物理和谐波分析联系起来。朗兰兹猜想的p进版本，包含了质数幂的可整除性，在朗兰兹的原始猜想上取得了令人兴奋的进展，并因此得到了一系列经典数论问题，如费马大定理。朗兰兹的猜想在其他情况下的进一步类似刺激了表征理论、代数几何和现代物理学的活动。最近，一些数学家为这些不同版本的朗兰兹猜想提出了一个统一的分类框架。本项目旨在通过对伽罗瓦表征空间的建模，通过范畴框架研究朗兰兹猜想的p进版本。伽罗瓦表示空间的几何是一个新的和丰富的研究方向，为培养年轻的研究人员和连接数论和几何表示理论的社区提供了机会。PI还计划通过外展方式招募和培训研究人员，方法是传播针对初级研究生的易获取的数论背景材料，以及组织地方和区域研讨会和会议。这个项目的目标是研究假设模p和p进朗兰兹对应，这些对应关系到伽罗瓦表示和p进约化群的模和积分表示。这种对应在推测上是由伽罗瓦表示法堆栈上的一束束来中介的。首席研究员将构建伽罗瓦表示堆栈的局部模型来分析这些猜想束，并使用Taylor-Wiles方法回答有关模块化提升，Breuil-Mezard猜想，Serre猜想的权重部分以及局部对称空间的模p上同调的问题。这些调查将允许应用和从模块化和几何表示理论。这个项目让本科生和研究生有机会通过组合方法和计算机代数包来探索局部模型。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。