An upper bound on the Abbes–Saito filtration for finite flat group schemes and applications

Tian, Yichao

doi:10.2140/ant.2012.6.231

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An upper bound on the Abbes–Saito filtration for finite flat group schemes and applications

Yichao Tian

Vol. 6 (2012), No. 2, 231–242

DOI: 10.2140/ant.2012.6.231

Abstract

Let $O_{K}$ be a complete discrete valuation ring of residue characteristic $p > 0$ , and $G$ be a finite flat group scheme over $O_{K}$ of order a power of $p$ . We prove in this paper that the Abbes–Saito filtration of $G$ is bounded by a linear function of the degree of $G$ . Assume $O_{K}$ has generic characteristic $0$ and the residue field of $O_{K}$ is perfect. Fargues constructed the higher level canonical subgroups for a “near from being ordinary” Barsotti–Tate group $G$ over $O_{K}$ . As an application of our bound, we prove that the canonical subgroup of $G$ of level $n \geq 2$ constructed by Fargues appears in the Abbes–Saito filtration of the $p^{n}$ -torsion subgroup of $G$ .

Keywords

finite flat group schemes, ramification filtration, canonical subgroups

Mathematical Subject Classification 2000

Primary: 14L15

Secondary: 14G22, 11S15

Milestones

Received: 3 May 2010

Revised: 2 May 2011

Accepted: 30 May 2011

Published: 24 June 2012

Authors

Yichao Tian
Mathematics Department Fine Hall Washington Road Princeton, NJ 08544 United States	Morningside Center of Mathematics 55 Zhong Guan Cun East Road Haidian District Beijing, 100190 China

Vol. 6, No. 2, 2012