Leaves in parahoric reductions of Shimura varieties

志村品种的叶片减少

• 批准号: 317095975
• 负责人: Professor Dr. Torsten Wedhorn
• 金额: --
• 依托单位: Arbeitsgruppe Algebra
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2018-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/317095975?language=en
• 关键词: Leaves; parahoric; reductions; Shimura; varieties

One of the central programs in Arithmetic Geometry is the web of conjectures and results predicted by the Langlands philosophy. In the last decade at almost every ICM a fields medal was awarded for work in this area (2002 Laurent Lafforgue, 2010 Ngo Bao Chau, 2014 Manjul Bhargava). The long-standing expectations that reductions of Shimura varieties are an essential tool in approaching the problems within the Langlands program has been impressively confirmed at the latest when Harris and Taylor proved the local Langlands conjecture at the end of the 1990's. Since then reductions of Shimura varieties and variants thereof have been a central tool for several substantial progresses (such as the work of Fargues, Boyer, Shin, or Scholze). In recent years there has also been a rapid development of tools in the field. Examples are the construction of reductions of very general classes of Shimura varities (e.g., by Vasiu, Kisin, and Pappas), the study of several important stratifications of general classes of reductions (e.g., by Haines, Görtz, Kottwitz, Rapoport, Viehmann, Wedhorn, Wortmann, Hamacher), or the construction of automorphic invariants (e.g., by Nicole, Goldring, Boxer, Koskivirta, Wedhorn and for perfectoizations of Shimura varieties by Caraiani and Scholze).One of the finest invariants is the isomorphism class of the $p$-divisible group together with its additional structure attached to a point in the reduction of the Shimura variety. The subsets where this invariant is constant are called leaves. In special cases much is known about these subsets (e.g. for moduli spaces of polarized abelian varieties by the work of Oort) but in full generality the structure of the leaves is still an open problem in particular in bad reduction. In this project the leaves will be studied in the very general case of parahoric reductions of Shimura varieties of abelian type.

算术几何中的一个中心程序是朗兰兹哲学所预测的图形和结果的网络。在过去的十年中，几乎每一届国际数学家大会都为这一领域的工作颁发了一个领域奖章（2002年Laurent Lafforgue，2010年Ngo Bao Chau，2014年Manjul Bhargava）。长期以来的期望，减少志村品种是一个重要的工具，在处理问题的朗兰兹计划已令人印象深刻的证实，最新的时候，哈里斯和泰勒证明了当地朗兰兹猜想在20世纪90年代末。从那时起，志村变种及其变种的简化一直是几个重大进展的核心工具（如法尔盖、博耶、申或肖尔泽的工作）。近年来，该领域的工具也有了迅速发展。例子是Shimura变种的非常一般的类的约简的构造（例如，由Vasiu，Kisin和Pappas），对一般归约类的几个重要分层的研究（例如，由Haines，Görtz，Kottwitz，Rapoport，Viehmann，Wedhorn，Wortmann，Hamacher），或者自守不变量的构造（例如，Nicole，Goldring，Boxer，Koskivirta，Wedhorn和Caraiani和Scholze对Shimura品种的完美化）。最好的不变量之一是$p$-整除群的同构类及其附加结构附加到Shimura品种的约简中的一个点上。这个不变量为常数的子集称为叶子。在特殊情况下，许多是已知的这些子集（例如模空间的极化阿贝尔品种的工作奥尔特），但在充分的一般性结构的叶子仍然是一个开放的问题，特别是在坏的减少。在这个项目中，叶将研究在非常普遍的情况下parahoric减少志村品种的阿贝尔型。