An Upper Bound for the Abbes-saito Filtration of Finite Flat Group Schemes and Applications

Let OK be a complete discrete valuation ring of residue characteristic p > 0, and G be a finite flat group scheme over OK 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 OK has generic characteristic 0 and the residue field of OK is perfect. Fargues constructed the higher level canonical subgroups for a “near from being ordinary” Barsotti-Tate group G over OK . As an application of our bound, we prove that the canonical subgroup of G of level n ≥ 2 constructed by Fargues appears in the AbbesSaito filtration of the p-torsion subgroup of G. Let OK be a complete discrete valuation ring with residue field k of characteristic p > 0 and fraction field K. We denote by vπ the valuation on K normalized by vπ(K) = Z. Let G be a finite and flat group scheme over OK of order a power of p such that G⊗K is étale. We denote by (Ga, a ∈ Q≥0) the Abbes-Saito filtration of G. This is a decreasing and separated filtration of G by finite and flat closed subgroup schemes. We refer the readers to [AS02, AS03, AM04] for a full discussion, and to section 1 for a brief review of this filtration. Let ωG be the module of invariant differentials of G. The generic étaleness of G implies that ωG is a torsion OK-module of finite type. There exist thus nonzero elements a1, · · · , ad ∈ OK such that