Shimura varieties and their local models

从1967年开始，朗兰兹提出了一个错综复杂的网络，将自同构形式理论和伽罗瓦表示理论联系起来。他还预测在志村品种及其局部模型的上同源群中也会编码这些对应关系。本研究旨在探讨这些猜想的几何含义，重点关注两个关键特征：全局猜想与局部猜想之间的兼容性以及功能原理。通过消除两个常见的限制性假设：适当性假设和P.E.L.型的Shimura型的子类限制，扩大了其在非定素数处的局部行为已被很好理解的Shimura型的类别。在第一种情况下，继alting - chai和Pink的工作之后，pi提出发展志村变量的算术紧化的积分理论。在第二个案例中，她计划按照Rapoport-Zink的策略，研究Hodge型的志村品种的坏还原。朗兰兹纲领提出了数论和调和分析中看似无关的对象之间存在一种关系。伽罗瓦群是存在于一个变量的给定多项式的根之间的所有对称性的集合，伽罗瓦表示是这种对称性如矩阵的实现。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令人难以置信的强大数学工具，将谐波分析定理转化为代数定理，反之亦然。在某些情况下，这种对应被编码在某些代数对象的几何中：志村变量和它们的局部模型。这个项目旨在揭示这些空间的几何形状如何反映出兰兰德猜想的一些最深刻的方面。

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.