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

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一些感兴趣的方程的整数或有理数解的数量。 (For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。虽然找到所有边长为有理数的直角三角形是很简单的，但可能令人惊讶的是，确定哪些整数可以是具有有理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编码在某些称为L函数的数学函数中。数学家罗伯特·朗兰兹（Robert Langlands）发展了一系列关于L-函数的理论（或数学预测），预测任何L-函数都应该产生于另一种称为自守形式的数学函数。研究自守形式和L-函数的一种方法是使用p-adic方法。这些方法涉及使用关于某个固定素数p的整除性质来研究自守形式和L-函数。最近，p-adic方法已经开始与朗兰兹的思想统一为所谓的“p-adic朗兰兹程序”。“这个项目的目的是在p-adic朗兰兹程序中开发新的结果和方法，主要是几何性质，并使用它们来建立朗兰兹几何的新实例。该奖项将支持学生在这一领域的研究，被认为是高兴趣的培训。该项目解决了以下基本问题：什么是潜在的几何结构相关的p-进伽罗瓦表示的mod p表示理论的p-进群？该项目建立在几个最近的发展，其中各种PI发挥了关键作用，包括建设的模栈参数化的p-adic局部领域的伽罗瓦群的p-adic表示和这些堆栈的局部模型，以及最近扩展的泰勒-怀尔斯修补方法，它涉及到研究相干层的局部模型，并导出代数几何。PI将研究的一些具体问题是潜在的晶体升降机的问题，一般的p-adic局部朗兰兹对应的建设，以及可能的局部性质的（先验的全球）修补constituent。 更一般地说，PI打算引入代数几何，分类，并导出的观点到p-adic朗兰兹计划，与获得新的见解，并在该领域的一些关键的开放问题上取得新的进展的意图。这个奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.