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

Abstract

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