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.

朗兰兹信件描述了两个不同的数学领域之间的联系：数论，包括对素数和多项式方程整数解的研究，以及调和分析，包括对光和声音如何分解为波的研究。例如，朗兰兹对应关系的某些实例将除多项式整数值的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是有用的，不仅可以使用经典几何，还可以使用从不寻常的尺寸和距离概念建立的替代几何理论，该理论检测被固定素数的整除性数字。这称为 p 进几何。 p 进几何中的基本形状看起来更像是分形，例如康托集，而不是我们在日常生活中在物理世界中遇到的形状，但尝试重新解释曲率等几何概念，以便它们仍然是富有成效的也可以用于 p-adic 世界。最近的完美类空间理论提供了 p 进几何的视角，非常适合研究朗兰兹对应中最重要的 p 进形状。该项目旨在将微积分的思想延续到完美类空间的研究中，以揭示朗兰兹对应的新结构特性，这最终将帮助我们理解有关整数和素数的基本问题。更准确地说，钻石理论（是完美类空间的商，通过非常好的等价关系）为 p 进几何提供了非常广泛的基础，其中包括大多数经典和现代感兴趣的对象，但在许多方面与拓扑流形理论比拓扑流形理论更相似复杂的分析空间。这项工作的目标是引入钻石解析结构的良好概念，然后应用该理论来研究朗兰兹对应中出现的 p 进自同构形式的表示理论方面。因此，特别强调理解 p 进空间的解析结构，这些空间是出现在朗兰兹对应的复几何中的复局部对称空间的通用覆盖的类似物。在复杂的设置中，分析结构可以直接在基座和通用盖之间传输，因为纤维是离散的，但在 p-adic 设置中，这受到纤维的有限拓扑和结构之间的不平凡的相互作用的阻碍。基础的刚性解析拓扑。该项目的一个重要的新见解是，在许多情况下，可以通过将刚性分析簇内的总空间嵌入为局部封闭的子菱形来局部理解这种相互作用。在某些情况下，这会通过有限路径的朴素概念产生巴拿赫-科尔梅斯切空间的新构造，并提出了完美性的自然标准，并具有对上同调消失的潜在应用。 PI 将分析具体例子，以阐明该分析理论的一般形式，同时还将 p 进自守形式理论、Shimura 簇的 p 进几何和 p 进数理论中一些最近和以前不相交的想法联系起来。 adic Langlands 通信。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。