Decompletion of cyclotomic perfectoid fields in positive characteristic

Berger, Laurent; Rozensztajn, Sandra

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

Soit $E$ un corps de caractéristique $p$. Le groupe ${\mathbf{Z}}_p^{\times }$ agit sur $E\left(\right(X\left)\right)$ par $a·f\left(X\right)=f({(1+X)}^a-1)$. Cette action s’étend à la complétion $X$-adique $\tilde{\mathbf{E}}$ de ${\cup }_{n\ge 0}E\left(\left(X^{1/p^n}\right)\right)$. Nous montrons comment récupérer $E\left(\right(X\left)\right)$ à partir du $E$-espace vectoriel valué $\tilde{\mathbf{E}}$ muni de son action de ${\mathbf{Z}}_p^{\times }$. Pour faire cela, nous introduisons la notion de vecteur super-Hölder dans certaines représentations $E$-linéaires de ${\mathbf{Z}}_p$. Ceci est un analogue en caractéristique $p$ de la notion de vecteur localement analytique dans les représentations de groupes de Lie $p$-adiques sur des Banach $p$-adiques.

Let $E$ be a field of characteristic $p$. The group ${\mathbf{Z}}_p^{\times }$ acts on $E\left(\right(X\left)\right)$ by $a·f\left(X\right)=f({(1+X)}^a-1)$. This action extends to the $X$-adic completion $\tilde{\mathbf{E}}$ of ${\cup }_{n\ge 0}E\left(\left(X^{1/p^n}\right)\right)$. We show how to recover $E\left(\right(X\left)\right)$ 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.

Classification : 11S80, 12J25, 13J05, 22E35

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