On some notions of good reduction for endomorphisms of the projective line

Abstract

Let Φ be an endomorphism of ${\mathbb{P}^1_{\overline{\mathbb{Q}}}}$, the projective line over the algebraic closure of ${\mathbb{Q}}$, of degree ≥ 2 defined over a number field K. Let v be a non-archimedean valuation of K. We say that Φ has critically good reduction at v if any pair of distinct ramification points of Φ do not collide under reduction modulo v and the same holds also for any pair of branch points. We say that Φ has simple good reduction at v if the map Φ _v , the reduction of Φ modulo v, has the same degree of Φ. We prove that if Φ has critically good reduction at v and the reduction map Φ _v is separable, then Φ has simple good reduction at v.