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。这一数学领域在密码学和物理学等领域也有应用。特别是分支数论考虑在这个项目是算术几何的利益在数论的性质进行了研究的几何方法。总体主题的研究是相互作用之间的算术几何和朗兰兹对应的号码领域。最近在这个领域有许多突破，例如证明互易性的新技术（模提升定理以及新兴的p-adic Langlands程序）和局部对称空间上同调中与挠类相关的伽罗瓦表示的构造。所有这些发展都依赖于能够p-基插值自守形式。p-adic自守形式的概念在Shimura簇的几何和上同调方面有一个自然的定义，因此p-adic算术几何对研究它们很有用。PI研究两个相互交织的领域：一方面，p-adic算术几何的应用（特别是perfectoid空间的理论），以p-adic自守形式，另一方面，p-adic和mod p类似的经典朗兰兹计划。具体来说，PI将通过泰勒-怀尔斯方法研究p-adic局部朗兰兹对应的新方法，进一步研究在志村簇的上同调中发生的扭转和相关伽罗瓦表示在p处的性质，并在志村簇的背景下开发一种新方法来解决泰特猜想的实例。PI打算使用的一些新技术是泰勒-怀尔斯修补方法，适用于完成上同调和匹配参数的局部变形环与Hecke算子。另一个关键的想法是通过相关的周期域来研究完全拟志村变种。该研究项目位于代数数论，表示论和代数几何的交叉点，重点是p-adic算术几何和数域的朗兰兹对应之间的相互作用。数论中的一个中心主题是数域的代数扩张的分类。类域理论解决了这一问题的扩展与阿贝尔伽罗瓦群。朗兰兹纲领提供了一个框架，将类场论广泛推广到非阿贝尔环境。其核心是自守表示和伽罗瓦表示之间的几何对应，这通常是由几何对象实现的，例如志村变种。因此，算术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例子，例如费马最后定理，佐藤-泰特猜想和塞尔猜想。

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.