Systèmes dynamiques non archimédiens et corps des norms ( Non-Archimedean Dynamic Systems and Fields of Norms )

Abstract

Let \(\mathcal{O}\)k be the ring of integers of a finite extension k of the field \(\mathbb{Q}\)p of p-adic numbers. The endomorphisms of a formal group law defined over \(\mathcal{O}\)k provide nontrivial examples of commuting formal series with coefficients in \(\mathcal{O}\)k. This article deals with the inverse problem formulated by Jonathan Lubin within the context of non-Archimedean dynamical systems. We present a large family of series, with coefficients in \(\mathbb{Z}\)p, which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over \(\mathbb{Q}\)p. We introduce the notion of minimally ramified series which turn out to be modulo p reductions of some series of this family. The commutant monoids of these minimally ramified series are determined by using the Fontaine–Wintenberger theory of the field of norms which allows an interpretation of them as automorphisms of \(\mathbb{Z}\)p-extensions of local fields of characteristic zero. A particularly effective example illustrating the paper is given by a family of series generalizing Cebysev polynomials