Arithmetic Aspects of the Langlands Program

数论是研究与整数性质有关的现象的数学分支。一个典型的数论问题是找出某个感兴趣的方程的所有整数解。通过综合这些方程图的拓扑和代数性质，我们得到一个被称为伽罗瓦表示的数学对象（数学数据的集合），它编码了原始方程的许多性质。数学家罗伯特·朗兰兹（Robert Langlands）提出了一系列的猜想（或数学预测），这些猜想预测了伽罗瓦表示与一些非常不同的群表示之间的密切关系，这些群表示被称为自同构表示，是由数学分析产生的。该项目的目标是通过将朗兰兹的猜想置于各种更结构化的几何框架中来取得进展，然后使用几何方法和视角来建立这两种不同类型的表征（伽罗瓦和自同构）之间的关系。该项目为研究生提供了培训机会。该研究项目的目标是调查算术朗兰兹程序的局部和全局方面。首席研究员Matthew Emerton博士将构建全局伽罗瓦表示的模栈，并研究它们的几何性质，以及将它们映射到局部伽罗瓦表示的模栈（乘积）的全局到局部限制映射的性质。这些研究与将Taylor- Wiles- Kisin修补方法扩展到剩余可约情况的问题以及对上同调的爱森斯坦部分的研究密切相关，Emerton博士将继续研究这些关系，特别是在奇数二维表示的背景下，这些关系与模曲线的上同调有关。Emerton博士还将开发一种新的方法来研究晶体局部-p伽罗瓦表示的模p变形理论，通过它们与棱柱形f晶体和量规的关系。由此产生的对这种变形理论的改进理解应该会导致新的全局结果，例如Serre猜想的权重部分的新情况和Breuil- Mezard猜想的新情况。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。

Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to find all the whole number solutions of some equation of interest. By synthesizing the topological and algebraic properties of the graphs of these equations, one obtains a mathematical object (a collection of mathematical data) called a Galois representation which encodes many of the properties of the originating equation. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) which anticipate a very close relationship between these Galois representations and some very different kinds of group representations, known as automorphic representations, which arise out of mathematical analysis. The goal of this project is to make progress on Langlands's conjectures by placing them in various more structural geometric frameworks, and then to use geometric methods and perspectives to establish relationships between these two different kinds of representations (Galois and automorphic). The project provides training opportunities for graduate students.The goal of the research project is to investigate both local and global aspects of the arithmetic Langlands program. The principal investigator, Dr. Matthew Emerton, will construct moduli stacks of global Galois representations, and study their geometric properties, as well as the properties of the global-to-local restriction maps which map them to (products of) moduli stacks of local Galois representations. These investigations are closely related to the problem of extending the Taylor--Wiles--Kisin patching method to the residually reducible case, and to the study of the Eisenstein part of cohomology, and Dr. Emerton will pursue these relationships, especially in the context of odd two-dimensional representations, which are connected to the cohomology of modular curves. Dr. Emerton will also develop a new approach to the mod p deformation theory of crystalline local-at-p Galois representations, via their relationship to prismatic F-crystals and gauges. The resulting improved understanding of this deformation theory should lead to new global results, such as new cases of the weight part of Serre's conjecture and new cases of the Breuil--Mezard conjecture.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.