Good Reduction for Endomorphisms of the Projective Line in Terms of the Branch Locus

TL;DR

The paper establishes a criterion for good reduction of endomorphisms of the projective line over number fields, generalizing previous results and exploring related notions of reduction and their preservation under iteration.

Contribution

It provides a new criterion for good reduction of endomorphisms, extending Zannier's result and connecting with simple and critically good reduction concepts.

Findings

Established a criterion for good reduction of endomorphisms.

Connected good reduction with simple and critically good reduction notions.

Characterized maps whose iterates preserve critically good reduction.

Abstract

Let $K$ be a number field and $v$ a non archimedean valuation on $K$. We say that an endomorphism $\Phi\colon \mathbb{P}_1\to \mathbb{P}_1$ has good reduction at $v$ if there exists a model $\Psi$ for $\Phi$ such that $\deg\Psi_v$, the degree of the reduction of $\Psi$ modulo $v$, equals $\deg\Psi$ and $\Psi_v$ is separable. We prove a criterion for good reduction that is the natural generalization of a result due to Zannier in \cite{Uz3}. Our result is in connection with other two notions of good reduction, the simple and the critically good reduction. The last part of our article is dedicated to prove a characterization of the maps whose iterates, in a certain sense, preserve the critically good reduction.