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: 1952667
• 负责人: Michael Harris
• 金额: $ 22.24万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1952667&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.

数字理论是研究与整数特性有关的现象的数学分支。一个典型的理论问题是确定某些感兴趣方程的整体或有理数解决方案的数量。 （例如，右三角形的三个侧面的长度与毕达哥拉斯定理相关。虽然很直接地找到所有正确的三角形的三角形是理性数字，但令人惊讶的是，确定哪些整体数字可以是对这些问题的答案。数学家罗伯特·兰兰兹（Robert Langlands）对L功能开发了一系列的猜想（或数学预测），这些命令预测，任何L功能都应来自另一种称为自动形态的数学功能。使用P-ADIC方法的使用自动形式和L功能的一种方法。这些方法涉及使用与某些固定质数P相对于一些固定质数P来研究自生物形式和L功能的方法。最近，P-ADIC方法已开始将Langlands的想法统一为所谓的“ P-Adic Langlands计划”。该项目旨在在P-ADIC Langlands计划（主要是几何性质）中开发新的结果和方法，并使用它们来建立兰兰兹猜想的新实例。该奖项将支持对这一研究领域的学生的培训。该项目解决以下基本问题：与P-ADIC群体的Mod P表示理论有关的基本几何结构是什么？该项目建立在各种PI都起着关键作用的几个最新发展的基础上，包括模量堆栈的构建参数化P-ADIC本地领域的Galois组和这些堆栈的本地模型的P-ADIC表示，以及最新的Taylor-Wiles修补方法的扩展，这些方法将其与局部模型的研究相关联。 PI会研究的一些具体问题是潜在的结晶升降机问题，P-Adic局部Langlands对应关系的构建以及（先验全球）补丁结构的局部性质。 更一般而言，PI打算将代数几何，分类和派生的观点引入P-Adic Langlands计划，目的是在领域中获得新的见解并在领域的某些关键开放问题上取得新的进步。该奖项在法定的任务中反映了NSF的法定任务，反映了通过评估范围的支持者，其知识范围众所周知。