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

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 $v$.Comment: 15 page

Topics: Mathematics - Number Theory, Mathematics - Algebraic Geometry, 14H25, 37P05, 37P35
Year: 2011
OAI identifier: oai:arXiv.org:1103.3853