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

Abstract

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 $n\geq 2$ constructed by Fargues appears in the Abbes-Saito filtration of the $p^n$-torsion subgroup of $\cG$.