The geometry of Shimura varieties at primes of bad reduction

本课题属于算术代数几何领域，主要研究阿贝尔变数和p可分群的模空间，特别是志村变数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纲领的框架内自然产生。拟议的调查侧重于两个不同但相互关联的问题。第一个是全局的，并致力于伽罗瓦表示的研究，有助于志村变种的上同调，重点是分枝素数。为此，PI在Kisin关于Hodge型的Shimura型的积分理论和Chai， Faltings和Pink关于PEL型的算术紧化的工作的基础上，提出将Shimura型及其算术紧化的积分理论扩展到包括坏约简素数。第二个问题是局部的，目的是确定给定p进群的哪些表示有助于志村变体的局部模型的上同调。特别是，PI计划通过结合局部和全局方法来寻找哈里斯猜想的新实例。朗兰兹的猜想探索了两个不同领域中看似无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。自同构形式是几个复变量的解析函数，具有许多自相似。伽罗瓦表示法是对单变量多项式方程的解之间存在的对称性的一种实现。朗兰兹最初的想法是通过一些其他的函数，称为l函数，来寻求这两个理论之间的联系。一方面，预测对应是用数论对象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算术空间，这些算术空间被期望在它们的几何实例中编码这些对应。

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.