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

TL;DR

This paper establishes a linear upper bound on the Abbes-Saito filtration for finite flat group schemes over certain valuation rings and demonstrates its application in identifying canonical subgroups within the filtration.

Contribution

It provides the first explicit linear bound on the Abbes-Saito filtration for finite flat group schemes and connects this bound to the existence of canonical subgroups in Barsotti-Tate groups.

Findings

The Abbes-Saito filtration is linearly bounded by the degree of the group scheme.

Canonical subgroups of level n are contained within the Abbes-Saito filtration.

The bound applies to both equal and mixed characteristic cases.

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 $\cG$.