The geometry of Shimura varieties at primes of bad reduction

不良还原素数时志村簇的几何形状

• 批准号: 1001077
• 负责人: Elena Mantovan
• 金额: $ 15.6万
• 依托单位: California Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-09-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001077&HistoricalAwards=false
• 关键词: geometry; Shimura; varieties; primes; bad

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.

这个项目是在算术代数几何领域，并关注于阿贝尔簇和p-可分群的模空间的研究，特别是与志村簇及其局部模型。 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。 拟议的调查集中在两个不同的，但相互关联的问题。 第一个是全球和追求的伽罗瓦表示志村品种专注于分歧素的上同调的研究。为此，PI建议扩展Shimura簇及其算术紧化的积分理论，以包括不良还原的素数，建立在Kisin对Hodge型Shimura簇的工作和Chai，Faltings和Pink对PEL型算术紧化的工作之上。 第二个问题是局部的，目的是确定一个给定的p-adic群的reprensetations有助于上同调的局部模型的志村品种。 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。朗兰兹的理论探索了两个不同领域中看似无关的对象之间的相互关系：调和分析中的自守形式和数论中的伽罗瓦表示。 自守形式是多个复变量的解析函数，具有许多自相似性。伽罗瓦表示是将一元多项式方程的解之间存在的对称性实现为矩阵。 朗兰兹最初的想法是通过一些其他的函数来寻求这两个理论之间的联系，这些函数被称为L-函数。一方面，预期对应是一种用数论方法组织分析对象的方法。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编码在这些对应的几何实例中。