## On quadratic rational maps with prescribed good reduction

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- by Clayton Petsche and Brian Stout
- Proc. Amer. Math. Soc.
**143**(2015), 1145-1158 - DOI: https://doi.org/10.1090/S0002-9939-2014-12291-X
- Published electronically: October 16, 2014
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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.## References

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## Bibliographic Information

**Clayton Petsche**- Affiliation: Department of Mathematics, Oregon State University, Corvallis, Oregon 97331
- Email: petschec@math.oregonstate.edu
**Brian Stout**- Affiliation: Ph.D. Program in Mathematics, CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
- Email: bstout@gc.cuny.edu
- 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
- © Copyright 2014
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**143**(2015), 1145-1158 - MSC (2010): Primary 37P45; Secondary 14G25, 37P15
- DOI: https://doi.org/10.1090/S0002-9939-2014-12291-X
- MathSciNet review: 3293730