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

Abstract

Let $Φ$ be an endomorphism of $\SR(\bar{\Q})$, the projective line over the algebraic closure of $\Q$, of degree $\geq2$ 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 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 $v$.