Shimura varieties and their local models

志村品种及其当地模型

• 批准号: 0701310
• 负责人: Elena Mantovan
• 金额: $ 12万
• 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: 2007
• 资助国家: 美国
• 起止时间: 2007-08-01 至 2010-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0701310&HistoricalAwards=false
• 关键词: Shimura; varieties; their; local; models

ABSTRACT for award DMS-0701310 of Mantovan Beginning in 1967 Langlands suggested an intricate web offar-reaching conjectures connecting the theories of automorphicforms and Galois representations. He also predicted some of thesecorrespondences to be encoded in the cohomology groups of Shimuravarieties and their local models. The proposed research pursues thestudy of the geometric implications of these conjectures, focusingon two key features: the compatibility between global and localconjectures and the functoriality principle. The PI aims to enlargethe class of Shimura varieties whose local behavior at unramifiedprimes is well understood by eliminating two common restrictivehypotheses: the assumption of properness and the restriction to thesubclass of Shimura varieties of P.E.L. type. In the first case, thePI proposes to develop the integral theory of arithmeticalcompactifications for Shimura varieties, following the works ofFaltings-Chai and Pink. In the second case, she plans to study thebad reduction of Shimura varieties of Hodge type, following thestrategy of Rapoport-Zink. The Langlands program proposes the existence of a relation betweenseemingly unrelated objects in number theory and harmonic analysis.A Galois group is the collection of all symmetries existing amongthe roots of a given polynomial in one variable and a Galoisrepresentation is the realization of such symmetries as matrices. Onthe other hand, automorphic forms are complex functions possessingmany self-similarities. A connection between the two theories as itis proposed by the Langlands program would provide an incrediblypowerful mathematical tool translating theorems of harmonic analysisinto theorems about algebra, and vice versa. In some cases, it isexpected such a correspondence to be encoded in the geometry ofcertain algebraic objects: the Shimura varieties and their localmodels. This project aims to shed some light on how the geometry ofthese spaces could reflect some of the deepest aspects of theLanglands' conjectures.

曼托万奖 DMS-0701310 摘要 从 1967 年开始，朗兰兹提出了一个复杂的、影响深远的猜想网络，将自同构理论和伽罗瓦表示联系起来。他还预测其中一些对应关系将被编码在志村品种的上同调群及其局部模型中。本研究致力于研究这些猜想的几何含义，重点关注两个关键特征：全局猜想和局部猜想之间的兼容性以及函子性原理。 PI 旨在通过消除两个常见的限制性假设来扩大 Shimura 品种的类别，这些品种在未分枝素数处的局部行为已被很好地理解：适当性假设和对 P.E.L 的 Shimura 品种子类的限制。类型。在第一种情况下，PI 提议继 Faltings-Chai 和 Pink 的工作之后，发展 Shimura 簇的算术紧致化积分理论。在第二个案例中，她计划按照Rapoport-Zink的策略，研究霍奇型志村品种的不良减少。朗兰兹纲领提出了数论和调和分析中看似不相关的对象之间存在一种关系。伽罗瓦群是一个变量中给定多项式的根之间存在的所有对称性的集合，伽罗瓦表示是矩阵等对称性的实现。另一方面，自同构形式是具有许多自相似性的复杂函数。朗兰兹纲领提出的两种理论之间的联系将提供一种极其强大的数学工具，将调和分析定理转化为代数定理，反之亦然。在某些情况下，期望这种对应关系被编码在某些代数对象的几何中：志村簇及其局部模型。该项目旨在阐明这些空间的几何形状如何反映朗兰兹猜想的一些最深层的方面。