Decompletion of cyclotomic perfectoid 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 ${\cup }_{n\ge 0}E((X^{1/p^n}))$. 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-Hölder 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.

v4: theorems now numbered in sections to match published version. v3: final version, accepted for publication in Annales Henri Lebesgue. Prop 3.2.9 and coro 3.2.10 are new. v2: added remarks 1.1.5 (an example), 1.2.7 (related work of Rodriguez Camargo), 1.3.4 (related work of Gulotta and Johansson-Newton) and 3.2.9 (exercise for the reader)