Geometric methods in the p-adic Langlands program

p 进朗兰兹纲领中的几何方法

基本信息

批准号：
2201112
负责人：
Sean Howe
金额：
$ 18万
依托单位：
University of Utah
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：
2022-08-15 至 2025-07-31
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201112&HistoricalAwards=false
关键词：
Geometric methods adic Langlands program

项目摘要

The Langlands correspondence describes a connection between two disparate areas of mathematics: number theory, which includes the study of prime numbers and integer solutions of polynomial equations, and harmonic analysis, which includes the study of how light and sound decompose into waves. For example, certain instances of the Langlands correspondence connect the prime numbers dividing integer values of a polynomial to vibrational frequencies of a very symmetric surface (like the fundamental tones of a musical instrument). For applications to number theory, it is useful to study these very symmetric surfaces and related higher dimensional shapes not only with classical geometry but also with an alternative theory of geometry built up from an unusual notion of size and distance that detects divisibility by a fixed prime number. This is called p-adic geometry. The basic shapes in p-adic geometry look more like fractals such as the Cantor set than like the shapes we encounter in our day to day lives in the physical world, but it is still fruitful to try to reinterpret geometric concepts like curvature so that they can be used also in the p-adic world. The recent theory of perfectoid spaces provides a perspective on p-adic geometry that is very well suited to studying the p-adic shapes that are most important in the Langlands correspondence. This project aims to carry over ideas from calculus to the study of perfectoid spaces in order to uncover new structural properties of the Langlands correspondence that will ultimately help us understand basic questions about the integers and prime numbers.More precisely, the theory of diamonds (which are quotients of perfectoid spaces by very nice equivalence relations) furnishes a very broad foundation for p-adic geometry that includes most classical and modern objects of interest but is in many ways more similar to the theory of topological manifolds than it is to the theory of complex analytic spaces. The goal of this work is to introduce a good notion of analytic structures on diamonds and then apply this theory to study representation theoretic aspects of p-adic automorphic forms as they arise in the Langlands correspondence. A special emphasis is thus put on understanding the analytic structure on the p-adic spaces which are analogs of the universal covers of complex locally symmetric spaces that appear in the complex geometry of the Langlands correspondence. In the complex setting the analytic structure can be transported directly between the base and the universal cover because the fibers are discrete, but in the p-adic setting this is obstructed by is a non-trivial interaction between the profinite topology of the fibers and the rigid analytic topology of the base. A crucial new insight in this project is that in many cases this interaction can be understood locally by embedding the total space inside of a rigid analytic variety as a locally closed subdiamond. This gives rise in some cases to a new construction of Banach-Colmez tangent spaces via a naive notion of profinite paths, and suggests a natural criterion for perfectoidness, with potential applications to cohomological vanishing. The PI will analyze concrete examples in order to elucidate the general shape of this analytic theory while also connecting some very recent and previously disjoint ideas in the theory of p-adic automorphic forms, the p-adic geometry of Shimura varieties, and the p-adic Langlands correspondence.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.

Langlands对应关系描述了数学的两个不同领域之间的联系：数字理论，其中包括对多项式方程的质量数和整数解的研究，以及谐波分析，其中包括对光和声音如何分解为WAV的研究。例如，Langlands对应关系的某些实例将分隔多项式的整数值连接到非常对称表面的振动频率（例如乐器的基本音调）。对于数字理论的应用，研究这些非常对称的表面和相关的更高维度不仅与经典的几何形状相关，而且还通过从不寻常的大小和距离概念中构建的替代几何理论来检测固定质量数字的分裂性。这称为P-Adic几何形状。 P-Adic几何形状中的基本形状看起来更像是分形的，例如Cantor套件，而不是我们在物理世界中日常生活中遇到的形状，但是尝试重新解释曲率等几何概念，以便在P-Adic世界中也可以使用它们。最新的完美体形空间理论提供了对P-ADIC几何形状的观点，非常适合研究Langlands对应中最重要的P-Adic形状。该项目旨在将思想从微积分到完美素的研究，以揭示兰兰德对应的新结构性特性，最终将帮助我们理解有关整数和质量数字的基本问题。更准确地说，更准确地说，更确切地说，是钻石的理论（非常良好的等价关系）为P-Ade coldication提供了一些类似的基础，但在许多范围内提供了类似的基础，并且在最多的基础上，以及最多的宗教级别，并在最多的基础上，在最多的层次中，它的几个阶层和质量是质量的，并且在最多的层次中，以及最多的层次。拓扑流形的理论比复杂的分析空间理论。这项工作的目的是在钻石上介绍一个很好的分析结构概念，然后将该理论应用于P-ADIC自多态形式的理论方面，因为它们在Langlands对应中出现。因此，特别重点是理解P-ADIC空间上的分析结构，这些分析结构是出现在兰兰兹对应关系复杂几何形状中的复杂局部对称空间的通用覆盖物的类似物。在复杂的环境中，由于纤维是离散的，因此可以直接在基础和通用覆盖率之间运输分析结构，但是在P-Adic设置中，这是由于纤维的涂料拓扑与碱基的刚性分析拓扑之间的非平凡相互作用。这个项目中的一个至关重要的新见解是，在许多情况下，可以通过将刚性分析品种内部的总空间嵌入本地封闭的子登录中，从而在本地理解这种相互作用。在某些情况下，这引起了Banach-Colmez切线空间的新结构，这是通过幼稚的路径的天真概念，并提出了完美素质的自然标准，并具有潜在的共同学消失的应用。 PI将分析具体示例，以阐明这种分析理论的一般形态，同时还将p-阿杜自动形式理论中的一些非常最新的，以前的脱节思想联系起来更广泛的影响审查标准。