Ramification Filtration and Differential Forms

Abstract

Let L be a complete discrete valuation field of prime characteristic p with finite residue field. Denote by Γ(v)L the ramification subgroups of ΓL=Gal(Lsep/L). We consider the category MΓLieL of finite Zp[ΓL]-modules H, satisfying some additional (Lie)-condition on the image of ΓL in AutZpH. In the paper it is proved that all information about the images of the groups Γ(v)L in AutZpH can be explicitly extracted from some differential forms Ω˜[N] on the Fontaine etale ϕ-module M(H) associated with H. The forms Ω˜[N] are completely determined by a canonical connection ∇ on M(H). In the case of fields L of mixed characteristic, which contain a primitive pth root of unity, we show that a similar problem for Fp[ΓL]-modules also admits a solution. In this case we use the field-of-norms functor to construct the corresponding ϕ-module together with the action of the Galois group of a cyclic extension L1 of L of degree p. Then our solution involves the characteristic p part (provided by the field-of-norms functor) and the condition for a “good” lift of a generator of Gal(L1/L). Apart from the above differential forms the statement of this condition uses the power series coming from the p-adic period of the formal group Gm.