Ramification of wild automorphisms of Laurent series fields

Abstract

Let K K be a complete discrete valuation field with residue class field k k , where both are of positive characteristic p p . Then the group of wild automorphisms of K K can be identified with the group under composition of formal power series over k k with no constant term and X X -coefficient 1 1 . Under the hypothesis that p > b 2 p > b^2 , we compute the first nontrivial coefficient of the p p th iterate of a power series over k k of the form f = X + ∑ i ≥ 1 a i X b + i f = X + \sum _{i \geq 1} a_iX^{b+i} . As a result, we obtain a necessary and sufficient condition for an automorphism to be “ b b -ramified”, having lower ramification numbers of the form i n ( f ) = b ( 1 + ⋯ + p n ) i_n(f) = b(1 + \cdots + p^n) . This is a vast generalization of Nordqvist’s 2017 theorem on 2 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 C p \mathbf {C}_p 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 2 -ramified power series.

Residue fixed point index and wildly ramified power series

“…This corollary solves [16, Conjecture 1.2] in the affirmative, for generic multiple fixed points of a fixed and small multiplicity, as well as [8, Conjecture 4.3]. In the case , Corollary 4 is [15, Main Theorem].…”

“…In the case of convergent power series, we also give an optimal lower bound for the distance to other periodic points (Theorem 3). This gives an affirmative solution to [16, Conjecture 1.2], for generic multiple fixed points of a fixed and small multiplicity, and to [8, Conjecture 4.3].…”

“…This last result also applies to the case , and asserts that a power series of the form () with is 1‐ramified if and only if In the case , Theorem 2 was shown by the first author [19, Theorem 1], with replaced by (). In the case and , Theorem 2 was shown by Kallal and Kirkpatrick in the first version of [8], with replaced by (). After a preliminary version of this paper was completed, we received a new version of [8] proving Theorem 2 when restricted to those satisfying , and with replaced by ().…”

“…In the case and , Theorem 2 was shown by Kallal and Kirkpatrick in the first version of [8], with replaced by (). After a preliminary version of this paper was completed, we received a new version of [8] proving Theorem 2 when restricted to those satisfying , and with replaced by ().…”

“…Such power series are called wildly ramified . See, for example, [9, 12, 23, 25] for background on wildly ramified power series, [8, 11, 14–16, 19, 21] for results related to this paper, and [6, 13, 17, 22] and references therein for local dynamics of analytic germs in positive characteristic. See also, for example, [4, 7] 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.…”