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On quadratic rational maps with prescribed good reduction

Authors: Clayton Petsche and Brian Stout
Journal: Proc. Amer. Math. Soc. 143 (2015), 1145-1158
MSC (2010): Primary 37P45; Secondary 14G25, 37P15
Published electronically: October 16, 2014
MathSciNet review: 3293730
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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 $ \phi :\mathbb{P}^1\to \mathbb{P}^1$ defined over $ K$ which have good reduction at all places outside $ S$ form a Zariski-dense subset of the moduli space $ \mathcal {M}_2$ 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 Zariski nondensity 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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Additional Information

Clayton Petsche
Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331

Brian Stout
Affiliation: Ph.D. Program in Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309

Keywords: Arithmetic dynamics, moduli spaces of rational maps, good reduction
Received by editor(s): February 14, 2013
Received by editor(s) in revised form: June 7, 2013
Published electronically: October 16, 2014
Additional Notes: The first author was supported by NSF grant DMS-0901147
The second author would like to thank Lucien Szpiro for his generous support under NSF grant DMS-0739346
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.