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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Uniform bounds for preperiodic points in families of twists


Authors: Alon Levy, Michelle Manes and Bianca Thompson
Journal: Proc. Amer. Math. Soc. 142 (2014), 3075-3088
MSC (2010): Primary 37P05; Secondary 11R99
Published electronically: May 19, 2014
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Abstract: Let $ \phi $ be a morphism of $ \mathbb{P}^N$ defined over a number field $ K.$ We prove that there is a bound $ B$ depending only on $ \phi $ such that every twist of $ \phi $ has no more than $ B$ $ K$-rational preperiodic points. (This result is analogous to a result of Silverman for abelian varieties.) For two specific families of quadratic rational maps over $ \mathbb{Q}$, we find the bound $ B$ explicitly.


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Additional Information

Alon Levy
Affiliation: Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T124
Email: levy@math.ubc.ca

Michelle Manes
Affiliation: Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822
Email: mmanes@math.hawaii.edu

Bianca Thompson
Affiliation: Department of Mathematics, University of Hawaii at Manoa, Honolulu, Hawaii 96822
Email: bat7@hawaii.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2014-12086-7
PII: S 0002-9939(2014)12086-7
Received by editor(s): May 8, 2012
Received by editor(s) in revised form: September 14, 2012
Published electronically: May 19, 2014
Additional Notes: The second and third authors’ work was partially supported by NSF-DMS 1102858.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society