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 $\cup_{n \geq 0} E((X^{1/p^n}))$. 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.Comment: 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