Decompletion of cyclotomic perfectoid fields in positive characteristic

TL;DR

This paper demonstrates how to recover a field of Laurent series over a characteristic p field from its completion with a group action, introducing super-Hölder vectors as an analogue of locally analytic vectors.

Contribution

It introduces the concept of super-Hölder vectors in characteristic p, enabling the recovery of the original field from its completed valued vector space with group action.

Findings

Recovery of $E((X))$ from $ ilde{ extbf{E}}$ using super-Hölder vectors

Introduction of super-Hölder vectors as an analogue to locally analytic vectors in characteristic p

Establishment of a method to analyze group actions on completed fields in positive characteristic

Abstract

Let $E$ be a field of characteristic $p$. The group $\mathbf{Z}_p^\times$ acts on $E((X))$ by $a \cdot f(X) = f((1+X)^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of $\cup_{n \geq 0} E((X^{1/p^n}))$. We show how to recover $E((X))$ from the valued $E$-vector space $\tilde{\mathbf{E}}$ endowed with its action of $\mathbf{Z}_p^\times$. To do this, we introduce the notion of super-H\"older vector in certain $E$-linear representations of $\mathbf{Z}_p$. This is a characteristic $p$ analogue of the notion of locally analytic vector in $p$-adic Banach representations of $p$-adic Lie groups.