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

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 >= 2 constructed by Fargues appears in the Abbes-Saito filtration of the p(n)-torsion subgroup of G.