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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Integral points and orbits of endomorphisms on the projective plane


Authors: Aaron Levin and Yu Yasufuku
Journal: Trans. Amer. Math. Soc. 371 (2019), 971-1002
MSC (2010): Primary 11G35, 14J20, 14R05, 14G40, 37P55
DOI: https://doi.org/10.1090/tran/7263
Published electronically: June 26, 2018
MathSciNet review: 3885168
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Abstract: We analyze when integral points on the complement of a finite union of curves in $ \mathbb{P}^2$ are potentially dense. When the logarithmic Kodaira dimension $ \bar {\kappa }$ is $ -\infty $, we completely characterize the potential density of integral points in terms of the number of irreducible components at infinity and the number of multiple members in a pencil naturally associated to the surface. When $ \bar {\kappa } = 0$, we prove that integral points are always potentially dense. The bulk of our analysis concerns the subtle case of $ \bar {\kappa }=1$. We determine the potential density of integral points in a number of cases by incorporating the structure theory of affine surfaces and developing an arithmetic framework for studying integral points on surfaces fibered over curves.

We also prove, assuming Lang-Vojta's conjecture, that an orbit under an endomorphism $ \phi $ of $ \mathbb{P}^2$ can contain a Zariski-dense set of integral points only if there is a nontrivial completely invariant proper Zariski-closed subset of $ \mathbb{P}^2$ under $ \phi $. This may be viewed as a generalization of a result of Silverman on  $ \mathbb{P}^1$.


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Additional Information

Aaron Levin
Affiliation: Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, Michigan 48824
Email: adlevin@math.msu.edu

Yu Yasufuku
Affiliation: Department of Mathematics, College of Science and Technology, Nihon University, 1-8-14 Kanda-Surugadai, Chiyoda-ku, 101-8308 Tokyo, Japan
Email: yasufuku@math.cst.nihon-u.ac.jp

DOI: https://doi.org/10.1090/tran/7263
Received by editor(s): January 27, 2017
Received by editor(s) in revised form: March 28, 2017
Published electronically: June 26, 2018
Additional Notes: The first author was supported in part by NSF grant DMS-1102563.
The second author was supported in part by JSPS Grant-in-Aid 15K17522 and by the Nihon University College of Science and Technology Grant-in-Aid for Fundamental Science Research.
Article copyright: © Copyright 2018 American Mathematical Society