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: 1952556
• 负责人: Brandon Levin
• 金额: $ 25.44万
• 依托单位: University of Arizona
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-01-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952556&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.

数论是研究与整数性质相关的现象的数学分支。一个典型的数论问题是确定某个感兴趣方程的整数解或有理数解的数量。 （例如，直角三角形的三条边的长度与毕达哥拉斯定理相关。虽然很容易找到所有边长为有理数的直角三角形，但令人惊讶的是，确定哪些整数可以是一个未解决的问题是有理边直角三角形的面积。）此类问题的答案通常可以用某些称为 L 函数的数学函数来编码。数学家罗伯特·朗兰兹 (Robert Langlands) 提出了一系列关于 L 函数的猜想（或数学预测），这些猜想预测任何 L 函数都应该源自另一种称为自守形式的数学函数。研究自同构形式和 L 函数的一种方法是使用 p-adic 方法。这些方法涉及使用某个固定素数 p 的整除性性质来研究自同构形式和 L 函数。最近，p-adic 方法开始与 Langlands 的思想统一为所谓的“p-adic Langlands 程序”。该项目旨在开发 p-adic Langlands 纲领中的新结果和方法（主要是几何性质），并使用它们来建立 Langlands 猜想的新实例。该奖项将支持对这一被认为高度感兴趣的研究领域的学生进行培训。该项目解决了以下基本问题：将 p-adic 伽罗瓦表示与 p-adic 的 mod p 表示理论联系起来的基本几何结构是什么团体？该项目建立在最近的几个发展基础上，其中各种 PI 发挥了关键作用，包括构建模堆栈，参数化 p-adic 局部域的 Galois 群的 p-adic 表示和这些堆栈的局部模型，以及最近的扩展泰勒-怀尔斯修补方法，将其与局部模型上的相干滑轮的研究以及派生的代数几何联系起来。 PI 将研究的一些具体问题是潜在的晶体升力问题、一般 p-adic 局部朗兰兹对应的构造以及（先验全局）修补构造的可能局部性质。 更一般地说，PI 打算将代数几何、分类和派生的观点引入到 p-adic Langlands 计划中，旨在对该领域的一些关键开放问题获得新的见解并取得新的进展。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。