Geometric methods in the p-adic Langlands program

朗兰兹的通信描述了两个不同的数学领域之间的联系：数论，它包括研究素数和多项式方程的整数解，以及调和分析，它包括研究光和声音如何分解成波。例如，朗兰兹对应的某些实例将除以多项式的整数值的素数与非常对称的表面的振动频率(如乐器的基音)联系在一起。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，不仅要用经典几何，而且要用另一种几何理论来研究，这种几何理论是从一个不寻常的大小和距离的概念建立起来的，它通过固定的素数来检测整除。这就是所谓的p-进几何。P元几何中的基本形状看起来更像像康托集这样的分形图，而不是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，以便它们也可以在p元世界中使用，仍然是卓有成效的。最近的完美拟态空间理论提供了一个关于p-进几何的视角，它非常适合于研究在朗兰兹对应中最重要的p-进形状。这个项目的目的是将微积分的思想延续到完美体空间的研究中，以揭示朗兰兹对应的新的结构性质，最终将帮助我们理解关于整数和素数的基本问题。更准确地说，钻石理论(通过非常好的等价关系得到的完美体空间的商)为p-进几何提供了非常广泛的基础，该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，而不是复杂分析空间的理论。这项工作的目的是在钻石上引入一个很好的解析结构的概念，然后应用这个理论来研究在朗兰兹对应中出现的p-进自同构形的表示理论方面。因此，一个特别的重点是理解p-进空间上的解析结构，它类似于复局部对称空间的泛覆盖，出现在朗兰兹对应的复几何中。在复杂的环境中，由于纤维是离散的，所以解析结构可以直接在底座和万能盖板之间传输，但在p-进的环境中，这被纤维的无限拓扑和底座的刚性解析拓扑之间的非平凡的相互作用所阻碍。这个项目的一个关键的新见解是，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分析种类中作为局部封闭的子钻石来局部理解。这在某些情况下产生了一种新的Banach-Colmez切空间的构造，它通过一种朴素的无限路径的概念，并提出了一个完美同态的自然准则，并潜在地应用于上同调零。PI将分析具体的例子，以阐明这一分析理论的一般形式，同时也将p-进自同构形式理论、下村变种的p-进几何以及p-进朗兰兹对应理论中的一些最新和以前不相关的观点联系起来。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。

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.