# Formal groups and lifts of the field of norms

## TL;DR

This paper investigates conditions under which the Galois action on the field of norms of a strictly APF extension can be lifted to characteristic zero, linking this to Lubin-Tate groups and $(phi,Gamma)$-modules.

## Contribution

It characterizes when such lifts exist, showing they are generated by torsion points of relative Lubin-Tate groups and describing the structure of the lifted Galois action.

## Key findings

- Lifts of the Galois action are generated by torsion points of relative Lubin-Tate groups.
- The power series for the lift are twists of semi-conjugates of endomorphisms.
- The work connects lifting the field of norms to the theory of $(phi,Gamma)$-modules.

## Abstract

Let $K$ be a finite extension of $\mathbf{Q}_p$. The field of norms of a strictly APF extension $K_\infty/K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{Gal}(K_\infty/K)$. When can we lift this action to characteristic zero, along with a compatible Frobenius map ? In this article, we explain what we mean by lifting the field of norms, explain its relevance to the theory of $(\varphi,\Gamma)$-modules, and show that under a certain assumption on the type of lift, such an extension is generated by the torsion points of a relative Lubin-Tate group and that the power series giving the lift of the action of the Galois group of $K_\infty/K$ are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.