Decompletion of cyclotomic perfectoid fields in positive characteristic

Berger, Laurent; Rozensztajn, Sandra

doi:10.5802/ahl.150

Berger, Laurent; Rozensztajn, Sandra

Decompletion of cyclotomic perfectoid fields in positive characteristic

Annales Henri Lebesgue, Volume 5 (2022), pp. 1261-1276.

Metadata

DOI10.5802/ahl.150

Abstract

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

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