Vol. 16, No. 2, 2022

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Formal groups and lifts of the field of norms

Léo Poyeton

Vol. 16 (2022), No. 2, 261–290
Abstract

Let K be a finite extension of p. The field of norms of a strictly APF extension KK is a local field of characteristic p equipped with an action of Gal (KK). 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 (φ,Γ)-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 KK are twists of semiconjugates of endomorphisms of the same relative Lubin–Tate group.

Keywords
field of norms, $(\phi, \Gamma)$-modules, formal groups, Lubin–Tate, archimedean dynamical systems
Mathematical Subject Classification 2010
Primary: 11S82
Secondary: 11S15, 11S20, 11S31, 13F25
Milestones
Received: 29 September 2019
Revised: 26 January 2021
Accepted: 17 June 2021
Published: 27 April 2022
Authors
Léo Poyeton
BICMR
Peking University
Haidian District
Beijing
China