FRG: Collaborative Research: Geometric Structures in the p-Adic Langlands Program

FRG：合作研究：p-Adic Langlands 计划中的几何结构

• 批准号: 2306369
• 负责人: Brandon Levin
• 金额: $ 25.44万
• 依托单位: William Marsh Rice University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-12-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306369&HistoricalAwards=false
• 关键词: FRG; Collaborative; Research; Geometric; Structures

Number theory is the branch of mathematics that studies phenomena related to properties of whole numbers. A typical number theoretic question is to determine the number of whole or rational number solutions of some equation of interest. (For example, the lengths of the three sides of a right triangle are related by the Pythagorean theorem. While it is straightforward to find all right triangles whose side lengths are rational numbers, it perhaps surprisingly remains an unsolved problem to determine which whole numbers can be the area of a right triangle with rational sides.) The answers to such questions can often be encoded in certain mathematical functions known as L-functions. The mathematician Robert Langlands has developed a series of conjectures (or mathematical predictions) regarding L-functions, which predict that any L-function should arise from another kind of mathematical function called an automorphic form. One approach to the study of automorphic forms and L-functions is the use of p-adic methods. These are methods that involve using divisibility properties with respect to some fixed prime number p to study automorphic forms and L-functions. Recently, p-adic methods have begun to be unified with Langlands's ideas into a so-called "p-adic Langlands program." This project aims to develop new results and methods in the p-adic Langlands program, primarily of a geometric nature, and to use them to establish new instances of Langlands's conjectures. The award will support the training of students in this area of research that is considered of high interest.This project addresses the following fundamental question: what are the underlying geometric structures relating p-adic Galois representations to the mod p representation theory of p-adic groups? The project builds on several recent developments in which the various PIs have played key roles, including the construction of moduli stacks parametrizing p-adic representations of the Galois groups of p-adic local fields and of local models for these stacks, and recent extensions of the Taylor-Wiles patching method which relate it to the study of coherent sheaves on the local models, and to derived algebraic geometry. Some specific questions that the PIs will study are the problem of potentially crystalline lifts, the construction of a general p-adic local Langlands correspondence, and the possible local nature of the (a priori global) patching constuction. More generally, the PIs intend to introduce algebro-geometric, categorical, and derived perspectives into the p-adic Langlands program, with the intention of gaining new insights into and making new progress on some of the key open problems in the field.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.

数论是数学的一个分支，研究与整数性质有关的现象。一个典型的数论问题是确定某个感兴趣的方程的整数解或有理数解的个数。(例如，直角三角形的三条边的长度由毕达哥拉斯定理联系在一起。虽然找到边长为有理数的所有直角三角形很简单，但可能令人惊讶的是，确定哪些整数可以是具有有理边数的直角三角形的面积仍然是一个悬而未决的问题。)这类问题的答案通常可以编码成称为L函数的某些数学函数。数学家罗伯特·朗兰兹发展了一系列关于L函数的猜想(或数学预测)，这些猜想(或数学预测)预言，任何L函数都应该由另一种称为自同构形的数学函数产生。研究自同构型和L函数的一种方法是使用p-进方法。这些方法涉及到利用关于某个固定素数p的可除性来研究自同构型和L函数。最近，p-adi方法已经开始与朗兰兹的思想相统一，形成了所谓的“p-adi朗兰兹计划”。这个项目的目的是在p-adic朗兰兹计划中发展新的结果和方法，主要是几何性质的，并用它们来建立朗兰兹猜想的新实例。该奖项将支持学生在这一被认为具有高度兴趣的研究领域的培训。这个项目解决了以下基本问题：p-进伽罗瓦表示与p-进群的mod p表示理论之间的潜在几何结构是什么？该项目建立在最近几个发展的基础上，在这些发展中，各种PI发挥了关键作用，包括构建模栈，该模栈将p-进局域的伽罗瓦群的p-进表示和这些堆的局部模型的P-进表示参数化，以及泰勒-威尔斯打补丁方法的最新扩展，该方法与局部模型上的相干层的研究有关，并与导出的代数几何有关。PI将研究的一些具体问题包括潜在结晶升力的问题，一般p-进局部朗兰兹对应的构造，以及(先验全局)修补构造的可能的局部性质。更广泛地说，PIS打算将代数几何、分类和派生的观点引入P-ADDIC朗兰兹计划，目的是对该领域的一些关键公开问题获得新的见解和取得新的进展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。