Potential density of projective varieties having an int-amplified endomorphism

TL;DR

This paper investigates the potential density of rational points on projective varieties with int-amplified endomorphisms, establishing results for non-uniruled and rationally connected varieties, and confirming potential density in dimensions up to three.

Contribution

It proves potential density for non-uniruled varieties with int-amplified endomorphisms and establishes the existence of dense rational orbits in rationally connected cases, extending results to dimension three.

Findings

Potential density holds for non-uniruled varieties with int-amplified endomorphisms.

Existence of a rational curve with a Zariski dense forward orbit in rationally connected varieties.

Potential density is confirmed for varieties of dimension up to three.

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