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

TL;DR

This paper investigates conditions under which endomorphisms of the projective line over number fields exhibit good reduction properties at non-archimedean valuations, linking critically good reduction with simple good reduction under separability.

Contribution

It establishes that critically good reduction combined with separability implies simple good reduction for endomorphisms of the projective line.

Findings

Critically good reduction and separability imply simple good reduction.

Defines and distinguishes between critically good and simple good reduction.

Provides conditions ensuring stable reduction behavior of endomorphisms.

Abstract

Let $\Phi$ 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 $\Phi$ has critically good reduction at $v$ if any pair of distinct ramification points of $\Phi$ do not collide under reduction modulo $v$ and the same holds for any pair of branch points. We say that $\Phi$ has simple good reduction at $v$ if the map $\Phi_v$, the reduction of $\Phi$ modulo $v$, has the same degree of $\Phi$. We prove that if $\Phi$ has critically good reduction at $v$ and the reduction map $\Phi_v$ is separable, then $\Phi$ has simple good reduction at $v$.