Geometric methods in the p-adic Langlands program

朗兰兹对应描述了数学两个不同领域之间的联系：数论，包括对素数和多项式方程的整数解的研究；谐波分析，包括对光和声音如何分解成波的研究。例如，朗兰兹对应的某些实例将多项式的整数值除以的素数与非常对称表面的振动频率（如乐器的基本音调）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，不仅用经典几何，而且用另一种几何理论，这种理论建立在一个不寻常的大小和距离的概念上，可以检测到一个固定素数的可整除性。这叫做p进几何。p进几何的基本形状看起来更像康托集合这样的分形，而不像我们在物理世界中日常生活中遇到的形状，但尝试重新解释曲率等几何概念仍然是富有成效的，这样它们也可以在p进世界中使用。最近的完美曲面空间理论为研究朗兰兹对应中最重要的p进几何提供了一个视角。该项目旨在将微积分的思想延续到完美曲面空间的研究中，以揭示朗兰兹对应的新结构性质，最终帮助我们理解关于整数和素数的基本问题。更准确地说，菱形理论（通过非常好的等价关系是完美空间的商）为p进几何提供了非常广泛的基础，p进几何包括大多数经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。本工作的目的是引入钻石解析结构的一个很好的概念，然后应用这个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进空间上的解析结构，p进空间类似于朗兰兹对应的复几何中出现的复局部对称空间的普遍覆盖。在复数情况下，由于纤维是离散的，解析结构可以直接在基和通用盖之间传递，但在p进情况下，这被纤维的有限拓扑和基的刚性解析拓扑之间的非平凡相互作用所阻碍。在这个项目中，一个至关重要的新见解是，在许多情况下，这种相互作用可以通过将总空间嵌入刚性解析变量内部作为局部封闭子钻石来局部理解。这在某些情况下产生了Banach-Colmez切空间的新构造，通过朴素的无限路径概念，并提出了完美性的自然准则，具有潜在的应用于上同调消失。PI将分析具体的例子，以阐明这一解析理论的一般形态，同时也将p进自同构形式理论、志村变元的p进几何和p进朗兰兹对应中的一些最近和以前不相关的思想联系起来。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。

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.