The geometry of Shimura varieties at primes of bad reduction

本项目在算术代数几何领域，主要研究交换簇和p-可除群的模空间，特别是Shimura簇及其局部模型的模空间。关于这些算术空间的几何和上同调的问题在朗兰兹计划的框架内自然产生。拟议的调查集中在两个截然不同但相互关联的问题上。第一个是全局的，致力于研究对Shimura变种上同调有贡献的Galois表示，重点是分支素数。为此，PI建议扩展Shimura变种及其算术紧化的积分理论，以包括不良约简的素数，建立在Hodge类型的Shimura变种的Kisin工作以及PEL型算术紧化的Chai、Faltings和Pink的工作的基础上。第二个问题是局部的，目的是确定给定的p-ady群的哪些表示有助于Shimura变种的局部模型的上同调。特别是，PI计划通过局部和全局方法的组合来寻找Harris猜想的新实例。朗兰兹的猜想探索了两个不同领域中看似不相关的物体之间的相互关系：调和分析中的自同构形式和数论中的伽罗瓦表示。自同构形是多个复变量的解析函数，具有许多自相似性。伽罗瓦表示是一元多项式方程的解之间的对称性作为矩阵的实现。朗兰兹最初的想法是通过另一种被称为L函数的函数来寻求这两种理论之间的联系。一方面，预期对应是根据数论来组织分析对象的一种方式。另一方面，它们的存在将为数论中许多悬而未决的问题提供答案。这个项目的目的是研究那些期望在这些对应的几何实例中编码的算术空间。

This project is in the field of arithmetic algebraic geometry, and is concerned with the study of moduli spaces of abelian varieties and p-divisible groups, and in particular with that of Shimura varieties and their local models. Questions about the geometry and cohomology of these arithmetic spaces arise naturally within the framework of the Langlands program. The proposed investigation focuses on two distinct but interralated problems. The first one is global and pursues the study of the Galois representations contributing to the cohomology of Shimura varieties focusing on ramified primes. T o do so, the PI proposes to extend the integral theory for Shimura varieties and their arithmetical compactifications to include primes of bad reduction, building on work of Kisin for Shimura varieties of Hodge type and on work of Chai, Faltings and Pink for arithmetical compactification of PEL type. The second problem is local and aims to identifying which reprensetations of a given p-adic group contribute to the cohomology of local models of Shimura varieties. In particular, the PI plans to purse new instances of a conjecture of Harris by a combination of local and global methods. Langlands' conjectures explore the interrelation between seemingly unrelated objects in two distinct fields: automorphic forms in harmonic analysis and Galois representations in number theory. An automorphic form is an analytic function of several complex variables, possessing many self-similarities. A Galois representation is a realizations of the symmetries existing among the solutions to a polynomial equation in one variable as matrices. Langlands' original idea was to pursue a connection between these two theories by the medium of some other functions, called L-functions. On one hand, the prospected correspondences are a way of organizing the analytic objects in terms of the number theoretic ones. On the other, their existence would provide answers to many open questions in number theory. This project is aimed to study those arithmetical spaces which are expected to encoded in their geometry instances of these correspondences.