Manuscripta Mathematica

, Volume 141, Issue 1–2, pp 315–331 | Cite as

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

  • Jung Kyu Canci
  • Giulio PeruginelliEmail author
  • Dajano Tossici


Let Φ be an endomorphism of \({\mathbb{P}^1_{\overline{\mathbb{Q}}}}\), the projective line over the algebraic closure of \({\mathbb{Q}}\), of degree ≥ 2 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 also 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.

Mathematics Subject Classification (2000)

14H25 37P05 37P35 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Jung Kyu Canci
    • 1
  • Giulio Peruginelli
    • 2
    Email author
  • Dajano Tossici
    • 3
  1. 1.Departement MathematicsUniversität BaselBaselSwitzerland
  2. 2.Institut für Analysis und Computational Number TheoryTechnische UniversitätGrazAustria
  3. 3.Scuola Normale SuperiorePisaItaly

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