Ramification of Wild Automorphisms of Laurent Series Fields

We study the structure of the Nottingham group N (Fp) of power series f ∈ X · Fp[[X]] with f (0) = 1 where p > 2 is prime, which is isomorphic to the group of wild automorphisms of Fp((X)). We concern ourselves with power series g ∈ N (Fp) with g 6= X for all m ≥ 1 (that is, power series of infinite order in N (Fp)), and we determine a necessary and sufficient criterion for such g ∈ N (Fp) having ramification type in(g) = 3(1 + · · · + p) given a finite number of coefficients of g. We also conjecture analogous results for higher ramification based on the results of a computer program, generalizing Fransson’s 2016 theorem on 2-ramified power series, as well as the work on minimally ramified power series used in a 2013 result of Lindahl and a 2015 result of Lindahl–Rivera-Letelier.