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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.

朗兰兹通信描述了两个不同的数学领域之间的联系：数论，包括研究素数和多项式方程的整数解，以及谐波分析，包括研究光和声音如何分解成波。例如，朗兰兹对应的某些例子将多项式的整数值除以素数与非常对称表面的振动频率（如乐器的基音）联系起来。对于数论的应用，研究这些非常对称的表面和相关的高维形状不仅用经典几何而且用另一种几何理论是有用的，这种几何理论是从一个不寻常的大小和距离概念建立起来的，它可以检测到被固定素数整除的可能性。这就是所谓的p-adic几何。p-adic几何中的基本形状看起来更像康托集这样的分形，而不像我们在日常生活中遇到的物理世界中的形状，但尝试重新解释曲率等几何概念，以便它们也可以用于p-adic世界，仍然是富有成效的。最近的完美空间理论提供了一个关于p-adic几何的视角，非常适合于研究在朗兰兹对应中最重要的p-adic形状。该项目旨在将微积分的思想带到完美空间的研究中，以揭示朗兰兹对应的新结构特性，最终帮助我们理解整数和素数的基本问题。更确切地说，钻石理论（它们是完美空间的等价关系）为p-包括大多数古典和现代感兴趣的对象，但在许多方面更类似于拓扑流形理论，而不是复解析空间理论的进几何。这项工作的目标是引入一个很好的概念，钻石上的分析结构，然后应用这一理论来研究表示理论方面的p进自守形式，因为它们出现在朗兰兹对应。一个特别强调的是，因此把理解的分析结构的p-adic空间是类似物的普遍覆盖复杂的局部对称空间，出现在复杂的几何朗兰兹对应。在复杂的设置中，由于纤维是离散的，因此分析结构可以直接在基底和通用覆盖之间传输，但是在p-adic设置中，这受到纤维的profinite拓扑和基底的刚性分析拓扑之间的非平凡相互作用的阻碍。在这个项目中一个重要的新见解是，在许多情况下，这种相互作用可以通过将整个空间嵌入刚性解析簇内部作为局部封闭的子菱形来局部理解。这在某些情况下产生了一个新的建设巴拿赫-科尔梅兹切空间通过一个天真的概念profinite道路，并提出了一个自然的标准perfectoidness，与潜在的应用上同调消失。PI将分析具体的例子，以阐明这一分析理论的一般形式，同时也连接一些非常近期和以前不相交的想法，在理论的p-adic自守形式，p-adic几何志村品种，和P-该奖项反映了NSF的法定使命，并被认为是值得通过使用基金会的智力价值和更广泛的评估支持影响审查标准。