Ramification of a finite flat group scheme over a local field
type: Item
instance of: scholarly article
title: Ramification of a finite flat group scheme over a local field
zbMATH Open document ID: 1107.14036
author: Shin Hattori
published in: Journal of Number Theory
publication date: 2006-11-15T00:00:00Z
review text: Let \(K\) be a complete discrete valuation field of mixed characteristic \((0,p)\), whose residue field need not be perfect, and \(\mathcal O_K\) its valuation ring. The author investigates ramification of a finite flat group scheme \(\mathcal G\) over \(\mathcal O_K\) following the results of \textit{A. Abbes} and \textit{T. Saito} [Am. J. Math. 124, 879--920 (2002; Zbl 1084.11064)], in particular the conductor \(c(\mathcal G)\) as introduced in Proposition 6.4 of that paper. Supposing that \(\mathcal G\) is killed by \(p^n\), it is proved that \[ c(\mathcal G) \leq ne + e/(p-1), \] where \(e\) is the absolute ramification index of \(K\). This result allows a generalization of a theorem of \textit{J.-M. Fontaine} [Invent. Math. 81, 515--538 (1985; Zbl 0612.14043)]. If \(\mathcal G\) is the \(p^n\)-torsion part of an elliptic curve, the exact value of \(c(\mathcal G)\) is given, depending on the elliptic curve being ordinary, fake ordinary, or truly supersingular.
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 14L15
Mathematics Subject Classification ID: 11S15
Mathematics Subject Classification ID: 11G07
zbMATH DE Number: 5072770
zbMATH Keywords: ramification filtration
zbMATH Keywords: Newton polygon
MaRDI profile type: Publication
full work available at URL: j.jnt.2005.08.006
OpenAlex ID: W1968663473
cites work: Canonical subgroups and p-adic vanishing cycles for abelian varieties
cites work: Ramification of local fields with imperfect residue fields. II
cites work: \(p\)-divisible groups, finite groups and filtered modules
cites work: There are no abelian varieties over \(\mathbb Z\).
cites work: Construction de représentations $p$-adiques
cites work: Schémas en groupes de type $(p,\ldots,p)$
cites work: Galois properties of points of finite order of elliptic curves
DOI: 10.1016/J.JNT.2005.08.006
description: scientific article; zbMATH DE number 5072770