Super-Hölder vectors and the field of norms

Berger, Laurent; Rozensztajn, Sandra

doi:10.2140/ant.2025.19.195

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Super-Hölder vectors and the field of norms

Laurent Berger and Sandra Rozensztajn

Vol. 19 (2025), No. 1, 195–211

DOI: 10.2140/ant.2025.19.195

Abstract

Let $E$ be a field of characteristic $p$ . In a previous paper of ours, we defined and studied super-Hölder vectors in certain $E$ -linear representations of $ℤ_{p}$ . In the present paper, we define and study super-Hölder vectors in certain $E$ -linear representations of a general $p$ -adic Lie group. We then consider certain $p$ -adic Lie extensions $K_{\infty} ∕ K$ of a $p$ -adic field $K$ , and compute the super-Hölder vectors in the tilt of $K_{\infty}$ . We show that these super-Hölder vectors are the perfection of the field of norms of $K_{\infty} ∕ K$ . By specializing to the case of a Lubin–Tate extension, we are able to recover $E ((Y))$ inside the $Y$ -adic completion of its perfection, seen as a valued $E$ -vector space endowed with the action of $𝒪_{K}^{\times}$ given by the endomorphisms of the corresponding Lubin–Tate group.

Keywords

Super-Hölder function, locally analytic function, field of norms, pro-$p$-group, perfectoid ring, decompletion, Lubin–Tate group, Mahler basis, $p$-adic wavelet

Mathematical Subject Classification

Primary: 11S15, 11S20, 11S80

Secondary: 11S31, 12J25, 13J05, 22E35

Milestones

Received: 16 March 2023

Accepted: 13 February 2024

Published: 4 December 2024

Authors

Laurent Berger
UMPA ENS de Lyon UMR 5669 du CNRS Lyon France

Sandra Rozensztajn
UMPA ENS de Lyon UMR 5669 du CNRS Lyon France

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Vol. 19, No. 1, 2025