Vol. 6, No. 2, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
An upper bound on the Abbes–Saito filtration for finite flat group schemes and applications

Yichao Tian

Vol. 6 (2012), No. 2, 231–242

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.

finite flat group schemes, ramification filtration, canonical subgroups
Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 14G22, 11S15
Received: 3 May 2010
Revised: 2 May 2011
Accepted: 30 May 2011
Published: 24 June 2012
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