Attaining Potentially Good Reduction in Arithmetic Dynamics

Abstract

Let K be a non-archimedean field, and let φ ∈ K(z) be a rational function of degree d ≥ 2. If φ has potentially good reduction, we give an upper bound, depending only on d, for the minimal degree of an extension L/K such that φ is conjugate over L to a map of good reduction. In particular, if d = 2 or d is less than the residue characteristic of K, the bound is d + 1. If K is discretely valued, we give examples to show that our bound is sharp. Fix the following notation throughout this paper.