Lifting the field of norms

Laurent Berger

doi:10.5802/jep.2

Abstract

Let K be a finite extension of Q_p. The field of norms of a p-adic Lie extension K_infty/K is a local field of characteristic p which comes equipped with an action of Gal(K_infty/K). When can we lift this action to characteristic 0, along with a compatible Frobenius map? In this note, we formulate precisely this question, explain its relevance to the theory of (phi,Gamma)-modules, and give a condition for the existence of certain types of lifts.

Highlights

Résumé (Relèvement du corps des normes). — Soit K une extension finie de Qp
Some preliminary computations suggest that a similar result may hold in certain cases if we assume that φq(T ) is an overconvergent power series in T
I do not know for which extensions we can expect the action of ΓK to be liftable in general

Summary

Lifting the field of norms

Let K be a finite extension of Qp and let K∞ be an infinite and totally ramified Galois extension of K that is “strictly arithmetically profinite” (see §1.2.1 of [Win83] for the definition, which we don’t use; arithmetically profinite means that the ramification subgroups ΓuK of ΓK are open and strictness is an additional condition). Note that if ΓK = Gal(K∞/K) is a p-adic Lie group, as recalled in §1.2.2 of [Win83], J.É.P. Let E be a finite extension of Qp such that kE = kK , let E be a uniformizer of E and let AK denote the E-adic completion of OE[[T ]][1/T ]. One reason for asking Question 1.2 is that it is relevant to the theory of (φ, Γ)-modules for OE-representations of GK This theory has been developed in [Fon90] when K∞ = K(μp∞ ), but it can be generalized to other extensions K∞/K for which the action of ΓK is liftable, as was observed for example in §2.1 of [Sch06]. These two facts imply that the functors of the theorem are mutually inverse

Embeddings into rings of periods

Lifts of finite height