Formal groups and lifts of the field of norms

DOI10.2140/ANT.2022.16.261arXiv1903.11106WikidataQ114045515 ScholiaQ114045515MaRDI QIDQ6316235

Léo Poyeton

Publication date: 26 March 2019

Abstract: Let $K$ be a finite extension of $mathbf{Q}_p$. The field of norms of a strictly APF extension $K_{i}nfty/K$ is a local field of characteristic $p$ equipped with an action of $mathrmGal(K_{i}nfty/K)$. 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 $K_{i}nfty/K$ are twists of semi-conjugates of endomorphisms of the same relative Lubin-Tate group.

Mathematics Subject Classification ID

Galois theory (11S20) Ramification and extension theory (11S15) Formal power series rings (13F25) Class field theory; (p)-adic formal groups (11S31) Non-Archimedean dynamical systems (11S82)

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