Decompletion of cyclotomic perfectoid fields in positive characteristic
Annales Henri Lebesgue, Volume 5 (2022), pp. 1261-1276.

### Abstract

Let $E$ be a field of characteristic $p$. The group ${\mathbf{Z}}_{p}^{×}$ acts on $E\left(\phantom{\rule{-0.166667em}{0ex}}\left(X\right)\phantom{\rule{-0.166667em}{0ex}}\right)$ by $a·f\left(X\right)=f\left({\left(1+X\right)}^{a}-1\right)$. This action extends to the $X$-adic completion $\stackrel{˜}{\mathbf{E}}$ of ${\cup }_{n\phantom{\rule{0.166667em}{0ex}}\ge \phantom{\rule{0.166667em}{0ex}}0}E\left(\phantom{\rule{-0.166667em}{0ex}}\left({X}^{1/{p}^{n}}\right)\phantom{\rule{-0.166667em}{0ex}}\right)$. We show how to recover $E\left(\phantom{\rule{-0.166667em}{0ex}}\left(X\right)\phantom{\rule{-0.166667em}{0ex}}\right)$ from the valued $E$-vector space $\stackrel{˜}{\mathbf{E}}$ endowed with its action of ${\mathbf{Z}}_{p}^{×}$. 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.

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