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涉及求解整数和密切相关的数系中的多项式方程。求解这些所谓的丢番图方程通常非常困难，而这种困难是加密算法的安全来源，例如，这些加密算法保证了互联网上的私人通信。此外，这些丢番图问题的丰富性与数学的其他分支有着深刻的联系。人们可以通过它的几何和动力学来卓有成效地研究这些方程的解的集合，将它们重新包装为伽罗瓦表示。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调和分析联系起来。朗兰兹猜想的p进版本包含了被素数的幂整除，这使得朗兰兹最初的猜想取得了令人兴奋的进展，从而导致了一系列经典的数论问题，如费马最后定理。朗兰兹猜想在其他背景下的进一步类似已经刺激了表象理论、代数几何和现代物理学的活动。最近，一些数学家为朗兰兹猜想的这些不同版本提出了一个统一的分类框架。本项目旨在通过对伽罗瓦表示的空间进行建模，通过范畴框架来研究P-进版本的朗兰兹猜想。伽罗瓦表示空间的几何是一个新的和丰富的研究方向，它为培养年轻的研究人员和联系数论和几何表示理论的社区提供了机会。国际数学家协会还计划通过外展、向研究生提供可获得的数论背景材料以及组织地方和地区研讨会和会议来招募和培训研究人员。该项目的目标是研究假想的mod p和p-adic朗兰兹对应关系，这些对应关系将伽罗瓦表示和p-adine约化群的模和积分表示联系起来。这种对应关系是由伽罗瓦表示的堆叠上的多个束来猜想地中介的。首席调查者将为Galois表示的堆栈构建局部模型，以分析这些猜想，并使用Taylor-Wiles方法回答有关模提升、Breuil-Mezard猜想、Serre猜想的权重部分以及局部对称空间的mod p上同调的问题。这些研究将允许模和几何表示理论的应用和来自模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包探索本地模型的机会。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。