Mod p geometry of Shimura varieties and applications

算术几何是数论的一个分支，主要研究多项式方程的整数解。虽然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，但解决方案往往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。这项建议涉及志村品种的研究;这些几何对象位于数论、代数几何和表示论之间的界面上，并且对算术几何中的一些重要和有影响的问题具有深远的应用。该提案旨在研究关于Shimura簇的模p几何的新问题和新现象，并将其应用于研究算术中的基本问题。主要的几何问题，我们专注于扩展以前的工作PI使用motivic上同调建立几何实现的Jacquet-Langlands对应，并研究几何的基本轨迹的Shimura品种的情况下，坏parahoric减少。然后，这些结果将被应用于研究新的情况下的Bloch-Kato猜想的特殊值的L-函数。另一部分是研究具有parahoric约化的交换型Shimura簇的半单局部zeta函数。该项目的最后一部分将使用志村品种回答一个基本问题，关于独立的伽罗瓦表示系统的l附加到阿贝尔品种。沿着，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.