In this paper we study lower ramification numbers of power series tangent to the identity that are defined over fields of positive characteristics p. Let g be such a series, then g has a fixed poin ...
The relation between lower ramification numbers and arithmetic dynamics over ultrametric fields is one of the motivations for this study. However, ramification numbers have also been considered in different contexts. In the study of the potential sequences of ramification numbers, Keating @cite_18 used the relation between the ramification numbers and abelian extensions of @math . Laubie and Sa " i ne @cite_14 @cite_7 could later improve these results by applying Wintenberger's theory on fields of norms @cite_5 . In @cite_6 the authors study Lubin's conjecture @cite_0 , on the relation between wildly ramified power series and formal groups.
@cite_18 @cite_14 @cite_7 @cite_6 @cite_0 @cite_5
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Let k be a field of characteristic p and let σ ∈ Autk k((t)) . For m ≥ 0 define im = vt(σpmt − t) − 1. We show that if i0 = 1 and i1 = 1 + bp with 0 < b < p − 1 then im = 1 + bp + bp2 + ctdot; + bpm for all m ≥ 1. Abstract Let k be a field of characteristic p and let γ ∈Aut k ( k (( t ))). For m ⩾0 define i m = v t ( γ p m t − t )−1. We show that if p ∤ i 0 is and i 1 p 2 − p +1) i 0 then there exists an integer b such that i m = i 0 + bp + bp 2 +…+ bp m for all m ⩾1. Let ( O ) k be the ring of integers of a finite extension k of the field ( Q ) p of p-adic numbers. The endomorphisms of a formal group law defined over ( O ) k provide nontrivial examples of commuting formal series with coefficients in ( 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 ( Z ) p , which satisfy Lubin's conjecture. These series are constructed with the help of Lubin–Tate formal group laws over ( 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 ( 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 Nous donnons une preuve que tout automorphisme sauvagement ramifie d'un corps de series formelles a une variable et a coefficients dans un corps parfait de caracteristique p provient de la construction du corps des nonnes d'une Zp-extension totalement ramifiee d'un corps local de caracteristique 0 ou p.