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Non-Archimedean dynamic systems and fields of norms. (Systèmes dynamiques non archimédiens et corps des norms.) (French) Zbl 1101.14057

Summary: 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 Chebyshev polynomials.

MSC:

14L05 Formal groups, \(p\)-divisible groups
11S31 Class field theory; \(p\)-adic formal groups
11S15 Ramification and extension theory
37F99 Dynamical systems over complex numbers
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