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.

代数被教授为求解方程的艺术，但事实上，很少有代数方程可以通过机械规则求解，尽管如此，人们通常可以通过其解集的形状来表征方程，即使解本身也是如此。朗兰兹对应关系源于这样的观察：在两个明显不同的数学领域——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，这些关系网允许问题的提出。有待解决的一个领域研究这些对应关系的数学分支称为自同构形式的理论，自同构形式的最简单的例子是三角学中熟悉的正弦和余弦函数，更一般的自同构形式是用几何来描述的。通过这种方式，自守形式的研究有助于数学所有分支的发展，在研讨会和会议上提出的当前项目中研究的问题将作为培训下一代专家的基础。它们还为哲学家和历史学家提供了当代数学特征的思想综合的例子，这对那些预测人工智能将在未来数学中发挥重要作用的人提出了特殊的挑战。对朗兰兹计划背景下的数字领域动机研究的贡献，延续了 PI 整个职业生涯中研究的核心主题。目前的提案分为两部分，特别是结合了动机方法。这Grothendieck-Deligne 权重理论​​ - 使用 Selberg 迹公式和表示理论来研究局部和全局朗兰兹对应（经典和 mod p）。特别是，概述了在局部场上归纳构造局部朗兰兹对应的策略。的正特征，并且对作者的广义拉马努金猜想方法的阿瑟参数进行了扩展，指出了相反的方向：它应用朗兰兹纲领和调和分析的见解来研究还原群。 (p-adic) 动机 L 函数，特别是平方根 p 进 L 函数，特别注意对可解释为上同调杯积的 Gan-Gross-Prasad 周期进行分类 与 Feng 和 联合的一个更具推测性的项目。 Mazur 旨在为 Iwasawa 理论背景下的 Venkatesh 动机猜想提供一个分类框架。该奖项反映了 NSF 的法定使命，并被认为值得支持通过使用基金会的智力优点和更广泛的影响审查标准进行评估。