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