The p-adic geometry of Shimura varieties and applications to the Langlands program

这是一个数论领域的研究项目。这一数学领域可以应用于密码学和物理学等领域。在这个项目中考虑的数论的特殊分支是算术几何，其中数论中感兴趣的性质是用几何方法研究的。研究的总体主题是算术几何和朗兰兹对应数域之间的相互作用。该领域最近取得了许多突破，例如证明互易性的新技术（模提升定理以及新兴的p进朗兰兹规划）和局部对称空间上同调中与扭转类相关的伽罗瓦表示的构造。所有这些发展都依赖于p基内插自同构形式的能力。p进自同构形式的概念在Shimura变异的几何和上同调中有一个自然的定义，因此p进算术几何对它们的研究是有用的。PI研究了两个相互交织的领域：一方面，p进算术几何（特别是完美空间理论）在p进自同构形式中的应用，另一方面，经典朗兰兹规划的p进和模p类似物。具体而言，PI将通过Taylor-Wiles方法研究p进局部朗兰兹对应的新方法，进一步研究Shimura变体上同调中的扭转以及相关伽罗瓦表示的p处性质，并开发Shimura变体背景下Tate猜想实例的新方法。PI打算使用的一些新技术是将Taylor-Wiles补片法应用于完全上同，并用Hecke算子匹配局部变形环中的参数。另一个关键思想是通过其相关的时期域来研究完美样志村品种。本研究项目是代数数论、表示论和代数几何的交叉领域，重点研究p进算术几何与数域的朗兰兹对应之间的相互作用。数论的一个中心主题是数域的代数扩展的分类。类场论在阿贝尔伽罗瓦群的扩展中解决了这个问题。朗兰兹纲领为将类场论广泛推广到非阿贝尔环境提供了一个框架。其核心是自同构表示和伽罗瓦表示之间的推测对应关系，这通常是通过几何对象实现的，例如志村变体。因此，等差几何为研究朗兰兹对应提供了许多重要的工具。最近数论中许多最引人注目的结果都是朗兰兹对应的实例，如费马大定理、佐藤-塔特猜想和塞尔猜想。

This is a research project in the general area of number theory. This area of mathematics has applications to areas such as cryptography and to physics. The particular branch of number theory considered in this project is arithmetic geometry where properties of interest in number theory are studied by geometric methods. The overall theme of the research is the interplay between arithmetic geometry and the Langlands correspondence for number fields. There have been many recent breakthroughs in the field, such as new techniques for proving reciprocity (modularity lifting theorems as well as the emerging p-adic Langlands program) and the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces. All of these developments have depended crucially on being able to p-adically interpolate automorphic forms. The concept of p-adic automorphic forms has a natural definition in terms of the geometry and cohomology of Shimura varieties and therefore p-adic arithmetic geometry is useful for studying them.The PI investigates two intertwined areas: on one hand, applications of p-adic arithmetic geometry (specifically the theory of perfectoid spaces) to p-adic automorphic forms and, on the other hand, p-adic and mod p analogues of the classical Langlands program. Specifically, the PI will study a new approach to the p-adic local Langlands correspondence via the Taylor-Wiles method, to further study torsion occurring in the cohomology of Shimura varieties and the properties at p of the associated Galois representations and to develop a new approach to instances of the Tate conjecture in the context of Shimura varieties. Some of the new techniques that the PI intends to use are the Taylor-Wiles patching method applied to completed cohomology and matching parameters in local deformation rings with Hecke operators. Another key idea involves studying perfectoid Shimura varieties via their associated period domains. The research project lies at the intersection of algebraic number theory, representation theory and algebraic geometry, with a focus on the interplay between p-adic arithmetic geometry and the Langlands correspondence for number fields. A central motif in number theory is the classification of algebraic extensions of number fields. Class field theory addresses this for extensions with abelian Galois group. The Langlands program provides a framework for a vast generalization of class field theory to the non-abelian setting. At its heart is the conjectural correspondence between automorphic representations and Galois representations, which is often realized by geometric objects, such as Shimura varieties. Therefore, arithmetic geometry provides many important tools for studying Langlands correspondences. Many of the most spectacular recent results in number theory are instances of the Langlands correspondence, such as Fermat's last theorem, the Sato-Tate conjecture and Serre's conjecture.