Remarks on endomorphisms and rational points

TL;DR

This paper explores how the behavior of rational self-maps near fixed points can be used to generate rational points on varieties, providing a simplified proof of potential density for lines on cubic fourfolds.

Contribution

It introduces a new perspective on using endomorphisms to study rational points and offers a simplified proof of a known density result for cubic fourfolds.

Findings

Behavior of rational self-maps near fixed points can produce rational points

Simplified proof of potential density on lines of cubic fourfolds

Method may extend to other varieties with similar structures

Abstract

Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).