Formal groups and lifts of the field of norms

Poyeton, Léo

doi:10.2140/ant.2022.16.261

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Formal groups and lifts of the field of norms

Léo Poyeton

Vol. 16 (2022), No. 2, 261–290

DOI: 10.2140/ant.2022.16.261

Abstract

Let $K$ be a finite extension of $ℚ_{p}$ . The field of norms of a strictly APF extension $K_{\infty} ∕ K$ is a local field of characteristic $p$ equipped with an action of $Gal (K_{\infty} ∕ K)$ . 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 $(φ, Γ)$ -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 $K_{\infty} ∕ K$ 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

Vol. 16, No. 2, 2022