Seminar Algebra and Geometry: Jung Kyu Canci (Universität Basel)

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

Let K be a number field and v a non archimedean valuation on K. We say that an endomorphism Φ : P1 → P1, defined over K, has good reduction at v if there exists a model Ψ for Φ such that deg Ψv, the degree of the reduction of Ψ modulo v, equals deg Ψ. We will present a criteria for the good reduction that is a generalization of a similar result due to Zannier where he considered some particular Belyi coverings (i.e. unramified except above{0,1,∞}). Our result applies to any rational maps and is in connection with other two notions of good reduction, the simple and the critically good reduction.

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