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 $p$th iterate of a power series over $k$ of the form $f = X + \sum_{i \geq 1} a_iX^{b+i}$. As a result, we obtain a necessary and sufficient condition for an automorphism to be ``$b$-ramified,'' having lower ramification numbers of the form $i_n(f) = b(1 + \cdots + p^n)$. This 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 $\mathbf{C}_p$ 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 $2$-ramified power series.