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

Abstract

Let $\varphi$ be an endomorphism of $\mathbb{P}^1_{\overline{\Q}}$ defined over a number field $K$. Given a discrete valuation $v$ of $K$, we consider here two notions of good reduction of $\varphi$ at $v$, called Standard Good Reduction (S.G.R., for short) and Critically Good Reduction (C.G.R.). If we consider the reduced map $\varphi_v$, in general its degree is smaller or equal to the degree of $\varphi$. We say that the map $\varphi$ has S.G.R. at $v$ if the degree of the reduced map $\varphi_v$ is equal to the degree of $\varphi$. This notion is frequently used in the study of arithmetical dynamical systems, allowing to reduce a global problem to a local problem. Another notion of good reduction has been recently introduced by Szpiro and Tucker to prove a finitess result about equivalence classes of endomorphisms of the projective line. We say that $\varphi$ has C.G.R. at $v$ if every pair of ramification points of $\varphi$ do not coincide modulo $v$ and the same holds for every pair of branch points. As an application of their result, Szpiro and Tucker showed that their theorem implies the well-known Shafarevich-Faltings theorem about the finiteness of the isomorphism classes of elliptic curves defined over a number field $K$ having good reduction outside a prescribed finite set of discrete valuations of $K$. Szpiro and Tucker already in their paper showed with same examples that these two notions are not equivalent. We prove here that if $\varphi$ has C.G.R. at $v$ and the reduced map $\varphi_v$ is separable, then $\varphi$ has S.G.R. at $v$.