Galois Representations and Automorphic Forms

伽罗瓦表示和自守形式

• 批准号: 1902265
• 负责人: Richard Taylor
• 金额: $ 43.64万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1902265&HistoricalAwards=false
• 关键词: Galois; Representations; Automorphic; Forms

The main purpose of this project is to allow the PI to continue to train graduate students in exciting, but very technically demanding, research in arithmetic geometry. This research concerns an extraordinary web of conjectures, including the Langlands conjectures, connecting algebra (i.e. the theory of polynomial equations) with analysis and geometry (particularly the study of geometric symmetries). Whenever these conjectures can be established, they provide a powerful tool which can reduce very hard problems in one domain to much easier problems in the other. The most celebrated example was Andrew Wiles' proof, after over 300 years of effort, of Fermat's last theorem. Wiles, partly in conjunction with the PI, reduced this algebraic problem to a much easier problem concerning symmetries of the hyperbolic plane (a sort of geometry popularized in Escher's "Circle Limit" woodcuts). The field of arithmetic geometry has seen extraordinary progress in recent decades. Among the many consequences flowing from this area are new error correcting codes which are essential for both modern computers (hard disks) and compact disks.In addition to supporting graduate students, the PI will continue his work on automorphy lifting theorems in the regular, but non-self-dual, case. Recently a group of 10 mathematicians, including the PI, made the first serious progress on this problem: They proved the potential automorphy of all elliptic curves over CM fields and the Ramanujan conjecture for the cohomology of Bianchi 3-manifolds. Further progress seems possible. One of the PI's students is also working on related issues and the PI will continue himself to think about these questions. In addition the PI will continue his attempts to prove that the eigenvalues of Hecke operators acting on algebraic, but non-regular, automorphic forms are algebraic.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.

该项目的主要目的是允许PI继续对研究生进行激动人心的算术几何学研究。这项研究涉及一个非凡的猜想网络，包括兰兰斯猜想，将代数（即多项式方程理论）与分析和几何形状相连（尤其是几何对称性的研究）。每当可以建立这些猜想时，它们就会提供一个强大的工具，可以将一个域中的非常困难的问题减少到另一个领域的问题更容易的问题。最著名的例子是安德鲁·威尔斯（Andrew Wiles）的证明，经过300多年的努力，Fermat的最后一个定理。威尔斯（Wiles）与PI结合了一部分，将这个代数问题减少到了一个关于双曲平面对称性的问题（一种在Escher的“圆圈极限”木刻中普及的几何形状）。最近几十年来，算术几何学领域取得了非凡的进步。从该领域流出的许多后果包括新的错误纠正代码，这对于现代计算机（硬盘）和紧凑型磁盘都是必不可少的。在支持研究生的情况下，PI将继续他在常规的，非自然的dual，case中以自动形态提升定理。最近，包括PI在内的10名数学家在这个问题上取得了第一个严重的进展：他们证明了CM领域的所有椭圆曲线的潜在自动形状，而Ramanujan猜想了Bianchi 3-manifolds的同胞。似乎进一步的进展似乎是可能的。 PI的一位学生也在研究相关问题，PI将继续考虑这些问题。此外，PI将继续他的尝试证明，Hecke运营商的特征值在代数方面行事，但不规则的，自动形式是代数。该奖项反映了NSF的法定任务，并通过使用该基金会的知识分子优点和广泛的影响来评估NSF的法定任务，并被认为是值得的。

项目成果

Potential automorphy over CM fields

CM 场上的潜在自同构

• DOI: 10.4007/annals.2023.197.3.2
• 发表时间: 2023
• 期刊: Annals of Mathematics
• 影响因子: 4.9
• 作者: Allen, Patrick;Calegari, Frank;Caraiani, Ana;Gee, Toby;Helm, David;Le Hung, Bao;Newton, James;Scholze, Peter;Taylor, Richard;Thorne, Jack
• 通讯作者: Thorne, Jack

Automorphy lifting with adequate image

具有足够图像的自同构提升

• DOI: 10.1017/fms.2023.3
• 发表时间: 2023
• 期刊: Sigma
• 作者: Miagkov, Konstantin;Thorne, Jack A.
• 通讯作者: Thorne, Jack A.

Potential automorphy for $$GL_n$$

$$GL_n$$ 的潜在自同构

• DOI: 10.1007/s00222-022-01161-6
• 发表时间: 2022
• 期刊: Inventiones mathematicae
• 影响因子: 3.1
• 作者: Qian, Lie