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New York Journal of Mathematics
Volume 28 (2022), 433-444

  

Jia Jia, Takahiro Shibata, and De-Qi Zhang

Potential density of projective varieties having an int-amplified endomorphism

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Published: February 23, 2022.
Keywords: Potential density, Int-amplified endomorphism, Arithmetic degree, Dynamical degree.
Subject: 37P55, 14G05, 14E30, 08A35.

Abstract
We consider the potential density of rational points on an algebraic variety defined over a number field K, i.e., the property that the set of rational points of X becomes Zariski dense after a finite field extension of K. For a non-uniruled projective variety with an int-amplified endomorphism, we show that it always satisfies potential density. When a rationally connected variety admits an int-amplified endomorphism, we prove that there exists some rational curve with a Zariski dense forward orbit, assuming the Zariski dense orbit conjecture in lower dimensions. As an application, we prove the potential density for projective varieties with int-amplified endomorphisms in dimension less than or equal to 3. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field.

Acknowledgements

The authors would like to thank the referee for very careful reading and the suggestions to improve the paper. The first, second and third authors are supported, from NUS, by the President's scholarship, a Research Fellowship and an ARF, respectively.


Author information

Jia Jia:
Department of Mathematics
National University of Singapore
Singapore 119076, Republic of Singapore

jia_jia@u.nus.edu

Takahiro Shibata:
Department of Mathematics
National University of Singapore
Singapore 119076, Republic of Singapore

mattash@nus.edu.sg

De-Qi Zhang:
Department of Mathematics
National University of Singapore
Singapore 119076, Republic of Singapore

matzdq@nus.edu.sg