On the Hodge-Newton filtration for \(p\)-divisible groups with additional structures
type: Item
instance of: scholarly article
zbMATH Open document ID: 1304.14057
author: Xu Shen
publication date: 2014-09-05T00:00:00Z
full work available at URL: https://arxiv.org/abs/1203.2541
Mathematics Subject Classification ID: 14L05
zbMATH DE Number: 6340381
zbMATH Keywords: \(p\)-divisible group
zbMATH Keywords: Newton polygon
zbMATH Keywords: Hodge polygon
zbMATH Keywords: Harder-Narasimhan filtration
zbMATH Keywords: Rapoport-Zink space
zbMATH Keywords: Harris' Conjecture
MaRDI profile type: Publication
OpenAlex ID: W2115989891
arXiv ID: 1203.2541
DOI: 10.1093/IMRN/RNT036
published in: IMRN. International Mathematics Research Notices
title: On the Hodge-Newton filtration for \(p\)-divisible groups with additional structures
review text: The major result of the subject paper shows that a \(p\)-divisible group with additional structures over a complete valuation ring of rank \(1\) with mixed characteristic admits a Hodge-Newton filtration assuming the Newton polygon and Hodge polygon of its special fiber has a nontrivial contact point that is also a break point for the Newton polygon.NEWLINENEWLINEMore specifically, let \(K/\mathbb{Q}_p\) be a complete discrete valuation field with perfect residue field \(k\). In the PEL symplectic case, a \(p\)-divisible group with additional structures over \(O_K\) is a triple \((H, \iota, \lambda)\) where \(H\) is a \(p\)-divisible group over \(O_K\), \(\iota : O_K \to \text{End}(H)\) is a homomorphism of algebras, \(\lambda : (H, \iota) \to (H^D, \iota^D)\) is a polarization. The following (HN) condition is assumed for all the major results of the paper: the Newton polygon \(\text{Newt}(H_k, \iota, \lambda)\) and the Hodge polygon \(\text{Hdg}(H_k, \iota, \lambda)\) possess a contact point \(x\) outside outside their extremal points, which is also a break point for the Newton polygon. Denote \(\hat{x}\) the symmetric point of \(x\). The main result of the paper is the following:NEWLINENEWLINE Theorem. Using the above notation, if \(x\) lies before \(\bar{x}\), then there are unique \(p\)-divisible subgroups with additional structures \((H_1, \iota) \subset (H_2, \iota) \subset (H, \iota)\) over \(O_K\) such that \(\lambda\) induces isomorphisms \((H_1, \iota) \cong ((H/H_2)^D, \iota')\) and \((H_2, \iota) \cong ((H/H_1)^D,\iota')\). The induced filtration \((H_{1k},\iota) \subset (H_{2k},\iota) \subset (H_k,\iota)\) over \(k\) is split. The Newton polygons (resp. Harder-Narasimhan polygons, Hodge polygons) of \((H_1, \iota)\), \((H_2/H_1, \iota)\), and \((H/H_2,\iota)\) are the parts of the Newton polygon (resp. Harder-Narasimhan polygon, Hodge polygon) of \((H, \iota,\lambda)\) up to \(x\), between \(x\) and \(\hat{x}\), and from \(\hat{x}\) on respectively.NEWL…
reviewed by: Xiao Xiao
description: scientific article; zbMATH DE number 6340381