FRG: Collaborative Research: Geometric Structures in the p-Adic 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确定某个感兴趣的方程的整数解或有理数解的个数。(例如，一个直角三角形的三条边的长度是由勾股定理联系起来的。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，确定哪些整数是有理数边的直角三角形的面积仍然是一个未解决的问题。这些问题的答案通常可以用某些称为l函数的数学函数进行编码。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（或数学预言），预言任何l函数都应该由另一种称为自同构形式的数学函数产生。研究自同构形式和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固定素数p的可整除性来研究自同构形式和l函数。最近，p进方法开始与朗兰兹的思想统一为所谓的“p进朗兰兹纲领”。该项目旨在开发p进朗兰兹程序的新结果和方法，主要是几何性质的，并使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。该项目解决了以下基本问题：p进伽罗瓦表示与p进群的模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，其中各种pi发挥了关键作用，包括模堆栈的构建，参数化p进局部域的伽罗瓦群的p进表示和这些堆栈的局部模型，以及Taylor-Wiles补丁方法的最近扩展，该方法将其与局部模型上的相干轴的研究联系起来，并导出代数几何。pi将研究的一些具体问题是潜在的结晶升降机问题，一般p进局部朗兰兹对应的构造，以及（先验的全局）修补构造的可能的局部性质。更一般地说，pi打算将代数几何、分类和衍生的观点引入p进朗兰兹程序，目的是在该领域的一些关键开放问题上获得新的见解和取得新的进展。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 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.