## Integral points and orbits of endomorphisms on the projective plane

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- by Aaron Levin and Yu Yasufuku PDF
- Trans. Amer. Math. Soc.
**371**(2019), 971-1002 Request permission

## 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
- MR Author ID: 775832
- 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
- MR Author ID: 681581
- Email: yasufuku@math.cst.nihon-u.ac.jp
- 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. - © Copyright 2018 American Mathematical Society
- 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
- MathSciNet review: 3885168