Leaves in parahoric reductions of Shimura varieties

算术几何的核心课程之一是朗兰兹哲学预测的猜想和结果的网络。在过去的十年里，几乎每一届国际管理委员会都因在这一领域的工作而授予菲尔兹奖章(2002年，劳伦特·洛夫格，2010年，Ngo Bao Chau，2014年，Manjul Bhargava)。长期以来，人们一直期望减少下村品种是解决朗兰兹计划中的问题的一个重要工具，这一期望最近得到了令人印象深刻的证实，哈里斯和泰勒在20世纪90年代末证明了当地的朗兰兹猜想。从那时起，下村品种及其变体的减少一直是几个实质性进展(如法格、博耶、辛或肖尔茨的工作)的中心工具。近年来，该领域的工具也有了快速的发展。例如，构造非常一般的Shimura变异类的约简(例如，Vasiu，Kisin和Pappas)，研究几个重要的一般约简类的分层(例如，Haines，Görtz，Kottwitz，Rapoport，Viehmann，Wedhorn，Wortmann，Hamacher)，或者构造自同构不变量(例如，Nicole，Goldring，Boxer，Koskivirta，Wedhorn，以及Caraiani和Scholze对Shimura变种的完美化简)。最好的不变量之一是p$-可分群的同构类及其附加结构，附加到Shimura变种的约化的一点上。这个不变量为常量的子集称为叶子。在特殊情况下，关于这些子集(例如，Oort的工作对于极化阿贝尔簇的模空间)已知很多，但总的来说，叶的结构仍然是一个开放的问题，特别是在坏的约化下。在这个项目中，我们将在非常普遍的情况下研究下村阿贝尔型变种的复指缩写。

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.