Decompletion of cyclotomic perfectoid fields in positive characteristic

2022/01/24 by Berger, Laurent, Rozensztajn, Sandra · 1 citation
#11S #12J #13J #22E #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2201.09688

Abstract

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 ∪_{n ≥ 0} E((X^1/pn)). 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.

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