Langlands correspondences and the arithmetic of automorphic forms

代数是作为解方程的艺术来教授的，但事实是，很少种类的代数方程可以用机械规则来解。然而，人们通常可以通过解集的形状来描述一个方程，即使解本身不能写下来。朗兰兹对应源于这样一种观察：出现在两个明显不同的数学领域——数论和数学物理的对称性——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被称为自同构形式理论。自同构形式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高维几何来描述的。这样，对自同构形式的研究有助于数学各个分支的发展。在讨论会和会议上提出的本项目所研究的问题将作为训练下一代专家的基础。它们还为哲学家和历史学家提供了当代数学特有的思想综合的例子，这对那些预测人工智能将在未来的数学中发挥重要作用的人提出了特殊的挑战。该项目是对朗兰兹计划背景下数字领域动机研究的贡献，延续了PI在其整个职业生涯中研究的核心主题。本提案分为两部分。第1部分结合了动机方法，特别是格罗滕迪克-德列涅权重理论将Selberg迹公式和表示理论应用于局部和全局朗兰兹对应的研究，包括经典的和模型的。特别地，概述了在局部正特征域上的局部朗兰兹对应的归纳构造策略，并将作者对广义拉马努金猜想的方法推广到Arthur参数。第2部分则指向相反的方向：它将Langlands规划和约化群的调和分析的见解应用于研究（p进）动机l函数，特别是平方根p进l函数，特别注意分类可以解释为上同调杯积的Gan-Gross-Prasad周期。与Feng和Mazur合作的一个更具推测性的项目，旨在为Venkatesh在Iwasawa理论背景下的动机猜想提供一个分类框架。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。

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.