#### Vol. 6, No. 2, 2012

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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
##### Abstract

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

##### Keywords
finite flat group schemes, ramification filtration, canonical subgroups
##### Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 14G22, 11S15
##### Milestones
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