Super-H\"older vectors and the field of norms

TL;DR

This paper extends the concept of super-H"older vectors from Z_p-representations to general p-adic Lie group representations, linking them to the perfection of the field of norms in p-adic Lie extensions, including Lubin-Tate cases.

Contribution

It introduces super-H"older vectors for p-adic Lie group representations and connects them to the field of norms' perfection in p-adic Lie extensions.

Findings

Super-H"older vectors in p-adic Lie extensions are the perfection of the field of norms.

In Lubin-Tate extensions, E((Y)) is recovered inside the Y-adic completion of its perfection.

The study generalizes super-H"older vectors to broader p-adic Lie group contexts.

Abstract

Let E be a field of characteristic p. In a previous paper of ours, we defined and studied super-H\"older vectors in certain E-linear representations of Z_p. In the present paper, we define and study super-H\"older 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-H\"older vectors in the tilt of K_\infty. We show that these super-H\"older 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.