Ramification of Wild Automorphisms of Laurent Series Fields

Abstract

Let K be a complete discrete valuation field with residue class field k, where both are of positive characteristic p. Then the group of wild automorphisms of K can be identified with the group under composition of formal power series over k with no constant term and X-coefficient 1. Under the hypothesis that p > b 2 , we compute the first nontrivial coefficient of the pth iterate of a power series over k of the form f = X + i≥1 a i X b+i . As a result, we obtain a necessary and sufficient condition for an automorphism to be "b-ramified," having lower ramification numbers of the form in(fThis is a vast generalization of Nordqvist's 2017 theorem on 2-ramified power series, as well as the analogous result for minimally ramified power series which proved to be useful for arithmetic dynamics in a 2013 paper of Lindahl on linearization discs in Cp and a 2015 result of Lindahl-Rivera-Letelier on optimal cycles over nonarchimedean fields of positive residue characteristic. The success of our computation is also promising progress towards a generalization of Lindahl-Nordqvist's 2018 theorem bounding the norm of periodic points of 2-ramified power series.

Residue fixed point index and wildly ramified power series

“…In the case of convergent power series, we also give an optimal lower bound for the distance to other periodic points (Theorem 3 in §1.3). This gives an affirmative solution to [LRL16b, Conjecture 1.2], for generic multiple fixed points of a fixed and small multiplicity, and to [KK16,Conjecture 4.3].…”

“…In the case q = 2, Theorem 2 was shown by the first named author [Nor17, Theorem 1], with résit(f ) replaced by (1.6). In the case q = 3 and K = F p , Theorem 2 was shown by Kallal and Kirkpatrick in the first version of [KK16], with résit(f ) replaced by (1.6). After a preliminary version of this paper was completed, we received a new version of [KK16] proving Theorem 2 when restricted to those q satisfying q 2 < p, and with résit(f ) replaced by (1.6).…”

“…wildly ramified power series, [KK16,LMS02,LN18,LRL16b,LRL16a,Nor17,RL03] for results related to this paper, and [HY83, Lin04, LZ10, Rug15] and references therein for local dynamics of analytic germs in positive characteristic. See also, e.g., [Joh88,Cam00] and references therein, for the myriad of group-theoretic results about the "Nottingham group", which is the group under composition formed by the wildly ramified power series.…”