Leaves in parahoric reductions of Shimura varieties

《算术几何》的核心课程之一是朗兰兹哲学所预测的猜想和结果的网络。在过去十年中，几乎每届ICM都为这一领域的工作颁发了田野奖（2002年Laurent Lafforgue， 2010年Ngo Bao Chau， 2014年Manjul Bhargava）。长期以来，人们一直期望志村品种的减少是解决朗兰兹计划中问题的重要工具，这一期望在哈里斯和泰勒在20世纪90年代末证明了局部朗兰兹猜想时得到了令人印象深刻的证实。从那以后，志村变体及其变体的简化成为了若干重大进展的核心工具（如法格斯、博耶尔、申或肖尔兹的工作）。近年来，该领域的工具也得到了迅速发展。例如，非常一般的Shimura变量类的约简的构造（例如，Vasiu， Kisin和Pappas），一般约简类的几个重要分层的研究（例如，Haines, Görtz, Kottwitz, Rapoport, Viehmann, Wedhorn, Wortmann, Hamacher），或自同构不变量的构造（例如，Nicole, Goldring, Boxer, Koskivirta， Wedhorn以及Caraiani和Scholze对Shimura变量的完美化）。最好的不变量之一是$p$可分群的同构类及其附加结构在Shimura变元的约简中附在一个点上。这个不变量为常数的子集称为叶。在一些特殊情况下，我们对这些子集已经有了很多了解（例如奥尔特对极化阿贝尔变体的模空间的研究），但总的来说，叶的结构仍然是一个开放的问题，特别是在坏约简中。在这个项目中，叶片将研究在非常一般的情况下，志村品种的阿贝尔型的旁消减。

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.