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 ${ℚ}_{p}$. The field of norms of a strictly APF extension ${K}_{\infty }∕K$ is a local field of characteristic $p$ equipped with an action of $\mathrm{Gal}\left({K}_{\infty }∕K\right)$. 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 $\left(\phi ,\Gamma \right)$-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.

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field of norms, $(\phi, \Gamma)$-modules, formal groups, Lubin–Tate, archimedean dynamical systems