# Good reduction for endomorphisms of the projective line in terms of the branch locus

Canci, Jung Kyu. (2015) Good reduction for endomorphisms of the projective line in terms of the branch locus. Preprints Fachbereich Mathematik, 2015 (36).

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Official URL: https://edoc.unibas.ch/69992/

Let K be a number field and v a non archimedean valuation on K. We say that an endomorphism $\Phi:\mathbb{P}_1 \to \mathbb{P}_1$ 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 [Z3]. 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.