Potential density of projective varieties having an int-amplified endomorphism

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 $\le 3$. We also study the existence of densely many rational points with the maximal arithmetic degree over a sufficiently large number field.

12 pages; comments are welcome!

References in corpus (5)

• Questions on self maps of algebraic varieties
• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• Invariant subvarieties with small dynamical degree
• Kawaguchi-Silverman conjecture for endomorphisms on rationally connected varieties admitting an int-amplified endomorphism
• Surjective endomorphisms of projective surfaces: the existence of infinitely many dense orbits