Mod p geometry of Shimura varieties and applications

算术几何是数论的一个分支，研究多项式方程的整数解。尽管该领域的激励问题相对容易陈述和理解，例如费马大定理的陈述，但其解决方案通常需要深刻的数学概念，并建立在与其他数学领域的深刻联系之上。本提案是关于志村品种的研究；它们是介于数论、代数几何和表示理论之间的几何对象，在算术几何的许多重要和有影响的问题上有着深远的应用。本文旨在研究志村变元模几何的新问题和新现象，并将其应用于基本的算术问题的研究。我们关注的主要几何问题是扩展PI先前的工作，即利用动机上同建立Jacquet-Langlands对应的几何实现，以及研究不良抛射约简情况下Shimura变量的基本轨迹的几何。这些结果将被应用于研究关于l函数特殊值的Bloch-Kato猜想的新情况。本课题的另一部分是研究具有旁约化的阿贝尔型Shimura变种的半简单局部zeta函数。项目的最后一部分将使用志村变量来回答一个关于伽罗瓦表示系统的独立性的基本问题，该系统附属于阿贝尔变量。在此过程中，我们的目标是开发新的工具和技术来研究这些问题。

Arithmetic geometry is a branch of number theory which is concerned with finding integer solutions to polynomial equations. Although the motivating problems in the field are relatively easy to state and understand, such as the statement of Fermat's Last Theorem, the solutions often require deep mathematical concepts and build on profound connections with other mathematical fields. This proposal is concerned with the study of Shimura varieties; these are geometric objects which lie on the interface between number theory, algebraic geometry and representation theory, and have far-reaching applications to a number of important and influential problems in arithmetic geometry.The proposal aims to study new problems and phenomena concerning the mod p geometry of Shimura varieties, and to apply these to study fundamental questions in arithmetic. The main geometric problems we focus on are to extend previous work of the PI on using motivic cohomology to establish geometric realizations of Jacquet-Langlands correspondences, and to study the geometry of the basic locus of Shimura varieties in the case of bad parahoric reduction. These results would then be applied to study new cases of the Bloch-Kato conjecture on special values of L-functions. Another part of the project is to study the semi-simple local zeta function of Shimura varieties of abelian type with parahoric reduction. The final part of the project will use Shimura varieties to answer a fundamental question concerning the independence of l of the system of Galois representations attached to abelian varieties. Along the way, we aim to develop new tools and techniques to study these questions.