A dynamical Shafarevich theorem for rational maps over number fields and function fields

2017/05/16 by Szpiro, Lucien, West, Lloyd
#14G25 #37F10 #37P45 #Algebraic Geometry (math.AG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1705.05489

Abstract

We prove a dynamical Shafarevich theorem on the finiteness of the set of isomorphism classes of rational maps with fixed degeneracies. More precisely, fix an integer d at least 2 and let K be either a number field or the function field of a curve X over a field k, where k is of characteristic zero or p>2d-2 that is either algebraically closed or finite. Let S be a finite set of places of K. We prove the finiteness of the set of isomorphism classes of rational maps over K with a natural kind of good reduction outside of S. We also prove auxiliary results on finiteness of reduced effective divisors in ℙ¹_K with good reduction outside of S and on the existence of global models for rational maps.

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