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 | References | Similar Articles | Additional Information

Abstract: We analyze when integral points on the complement of a finite union of curves in are potentially dense. When the logarithmic Kodaira dimension is , 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 , we prove that integral points are always potentially dense. The bulk of our analysis concerns the subtle case of . 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 of can contain a Zariski-dense set of integral points only if there is a nontrivial completely invariant proper Zariski-closed subset of under . This may be viewed as a generalization of a result of Silverman on .

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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.

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© Copyright 2018
American Mathematical Society