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

Let $cO_K$ be a complete discrete valuation ring of residue characteristic $p>0$, and $G$ be a finite flat group scheme over $cO_K$ of order a power of $p$. We prove in this paper that the Abbes-Saito filtration of $G$ is bounded by a simple linear function of the degree of $G$. Assume $cO_K$ has generic characteristic 0 and the residue field of $cO_K$ is perfect. Fargues constructed the higher level canonical subgroups for a Barsotti-Tate group $cG$ over $cO_K$ which is not too supersingular. As an application of our bound, we prove that the canonical subgroup of $cG$ of level $ngeq 2$ constructed by Fargues appears in the Abbes-Saito filtration of the $p^n$-torsion subgroup of $cG$.