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

Abstract

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 not too supersingular 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 Abbes-Saito filtration of the p n -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 vthe valuation on K normalized by v�(K × ) = Z. Let G be a finite and flat group scheme over OK of order a power of p such that G K is etale. We denote by (G a ;a 2 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 !G be the module of invariant differentials of G. The generic etaleness of G implies that !G is a torsion OK-module of finite type. There exist thus nonzero elements a1;���;ad 2 O K such that !G ' d