Cycles for rational maps with good reduction outside a prescribed set

2005/04/26 by J. K. Canci, Canci, J. K.
Mathematics · #11G99 (Primary) 14E05 (Secondary) #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #math.AG #math.NT #msc:11G99 #msc:14E05

paper · pdf · doi:10.48550/arxiv.math/0504533

30 pages, changed content

arxiv created 2006/07/17 · arxiv updated 2009/12/01

Abstract

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let R_S be the ring of S-integers of K. In the present paper we study the cycles for rational maps of ℙ₁(K) of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P₀,P₁,...,P_n-1) and (Q₀,Q₁,...,Q_n-1) of points of ℙ₁(K) are equivalent if there exists an automorphism A∈\rm PGL₂(R_S) such that P_i=A(Q_i) for every index i∈\0,1,...,n-1\. We prove that if we fix two points P₀,P₁∈ℙ₁(K), then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P₀,P₁ as consecutive points is finite and depends only on S. We also prove that this result is in a sense best possible.

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