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

Let E be a field of characteristic p. The group Z p × acts on E((X)) by a·f(X)=f((1+X) a -1). This action extends to the X-adic completion E ˜ of n0 E((X 1/p n )). We show how to recover E((X)) from the valued E-vector space E ˜ endowed with its action of Z p × . 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.

Soit E un corps de caractéristique p. Le groupe Z p × agit sur E((X)) par a·f(X)=f((1+X) a -1). Cette action s’étend à la complétion X-adique E ˜ de n0 E((X 1/p n )). Nous montrons comment récupérer E((X)) à partir du E-espace vectoriel valué E ˜ muni de son action de Z p × . Pour faire cela, nous introduisons la notion de vecteur super-Hölder dans certaines représentations E-linéaires de 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.

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DOI: 10.5802/ahl.150
Classification: 11S80, 12J25, 13J05, 22E35
Berger, Laurent 1; Rozensztajn, Sandra 1

1 UMPA de l’ÉNS de Lyon UMR 5669 du CNRS (France)
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Berger, Laurent; Rozensztajn, Sandra. Decompletion of cyclotomic perfectoid fields in positive characteristic. Annales Henri Lebesgue, Volume 5 (2022), pp. 1261-1276. doi : 10.5802/ahl.150. http://www.numdam.org/articles/10.5802/ahl.150/

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