Lifting the field of norms

TL;DR

This paper investigates conditions under which the Galois action on the field of norms in p-adic Lie extensions can be lifted to characteristic zero, with implications for (phi,Gamma)-modules theory.

Contribution

It formulates a precise question about lifting Galois actions from characteristic p to zero and provides a condition for such lifts to exist.

Findings

Provides a criterion for lifting Galois actions to characteristic zero.

Clarifies the relevance of lifts to the theory of (phi,Gamma)-modules.

Connects the lifting problem to the structure of p-adic Lie extensions.

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.