On quadratic rational maps with prescribed good reduction

2012/08/28 by Petsche, Clayton, Stout, Brian
#37P15 (Secondary) #37P45 (Primary) 14G25 #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1208.5794

Abstract

Given a number field K and a finite set S of places of K, the first main result of this paper shows that the quadratic rational maps ϕ:\mathbb P¹→\mathbb P¹ defined over K which have good reduction at all places outside S comprise a Zariski-dense subset of the moduli space \mathcal M₂ parametrizing all isomorphism classes of quadratic rational maps. We then consider quadratic rational maps with double unramified fixed-point structure, and our second main result establishes a geometric Shafarevich-type non-Zariski-density result for the set of such maps with good reduction outside S. We also prove a variation of this result for quadratic rational maps with unramified 2-cycle structure.

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