2021/05/25 by Abrashkin, Victor
#11S15 #11S20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2105.11968
Let L be a complete discrete valuation field of prime characteristic p with finite residue field. Denote by ΓL(v) the ramification subgroups of ΓL=Gal(Lsep/L). We consider the category MΓLLie 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 ramification subgroups ΓL(v) 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 connection ∇ on M(H). In the case of fields L of mixed characteristic containing a primitive p-th root of unity we show that the similar problem for \mathbbFp[ΓL]-modules also admits a solution. In this case we use the field-of-norms functor to construct the coresponding ϕ-module together with the action of a cyclic group of order p coming from a cyclic extension of L. Then the solution involves the characteristic p part (provided by the field-of-norms functor) and the condition for a "good" lift of a generator of the involved cyclic group of order p. Apart from the above differential forms the statement of this condition also uses a power series coming from the p-adic period of the formal group \mathbbGm.