Langlands correspondences and the arithmetic of automorphic forms

朗兰兹对应和自守形式的算术

• 批准号: 2302208
• 负责人: Michael Harris
• 金额: $ 31.74万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing 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=2302208&HistoricalAwards=false
• 关键词: Langlands; correspondences; arithmetic; automorphic; forms

Algebra is taught as the art of solving equations, but the fact is that very few kinds of algebraic equations can be solved by mechanical rules. Nevertheless, one can often characterize an equation by the shape of its set of solutions, even if the solutions themselves cannot be written down. The Langlands correspondences grow out of the observation that the shapes of the equations that arise in two apparently different areas of mathematics — number theory and the symmetries of mathematical physics — are linked by a complex web of relations that allow questions in one area to be solved by reference to the other area. The branch of mathematics that studies these correspondences is called the theory of automorphic forms. The simplest examples of automorphic forms are the familiar sine and cosine function from trigonometry. More general automorphic forms are described in terms of geometry in higher dimensions. In this way the study of automorphic forms contributes to the development of all branches of mathematics. Problems studied in the present project, presented at seminars and conferences, will serve as the basis for training the next generation of specialists. They also provide examples for philosophers and historians of the kind of synthesis of ideas that is characteristic of contemporary mathematics, and that presents a special challenge for those who prodict that artificial intelligence will play a prominent role in the mathematics of the future.This project is a contribution to the study of motives over number fields in the setting of the Langlands program, continuing a theme that has been central to the PI's research throughout his career. The present proposal is divided into two parts. Part 1 combines motivic methods, — especially the Grothendieck-Deligne theory of weights — with the Selberg trace formula and representation theory to the study of local and global Langlands correspondences, both classical and mod p. In particular, a strategy is outlined for an inductive construction of the local Langlands correspondence over local fields of positive characteristic, and an extension is proposed to Arthur parameters of the author's approach to the generalized Ramanujan conjecture. Part 2 points in the opposite direction: it applies the insights of the Langlands program and harmonic analysis on reductive groups to study (p-adic) motivic L-functions, especially square root p-adic L-functions, with special attention to classifying the Gan-Gross-Prasad periods that can be interpreted as cohomological cup products. A more speculative project, joint with Feng and Mazur, aims to provide a categorical framework for Venkatesh's motivic conjectures in the setting of Iwasawa theory.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.

代数学是作为解方程的艺术来教授的，但事实上，很少有代数方程能用机械法则来解。 然而，人们通常可以通过方程的解集的形状来描述方程的特征，即使解本身不能写下来。 朗兰兹的对应关系源于这样一种观察，即数学的两个明显不同的领域--数论和数学物理的对称性--中出现的方程的形状是由一个复杂的关系网联系在一起的，这个关系网允许一个领域中的问题通过参考另一个领域来解决。 研究这些对应关系的数学分支叫做自守形式论。 自守形式最简单的例子是三角函数中常见的正弦和余弦函数。 更一般的自守形式在更高维度的几何学中描述。 这样，自守形式的研究有助于数学所有分支的发展。 本项目中研究的问题在研讨会和会议上提出，将作为培训下一代专家的基础。 它们也为哲学家和历史学家提供了当代数学所特有的那种思想综合的例子，这对那些预言人工智能将在未来的数学中发挥突出作用的人来说是一个特殊的挑战。这个项目是对朗兰兹纲领背景下数域动机研究的贡献，继续一个主题，一直是中央PI的研究在他的职业生涯。 本提案分为两部分。 第1部分结合motivic方法，特别是Grothendieck-Deligne理论的重量-与Selberg迹公式和表示理论的研究，当地和全球朗兰兹对应，无论是经典的和mod p。 特别是，一个战略概述了归纳建设的本地Langlands对应的本地领域的正特征，并提出了一个扩展亚瑟参数的作者的方法推广Ramanujan猜想。 第二部分则是相反的方向：它应用朗兰兹纲领和调和分析对约化群的见解来研究（p-adic）动机L-函数，特别是平方根p-adic L-函数，并特别注意对可以解释为上同调杯积的Gan-Gross-Prasad周期进行分类。 一个更具投机性的项目，与冯和马祖尔联合，旨在提供一个明确的框架，为文卡特什的motivic在岩泽理论的设置。这个奖项反映了美国国家科学基金会的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。