### Abstract

Let X be an algebraic variety and let f:X→X be a rational self-map with a fixed point q, where everything is defined over a number field K. 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 proved by Claire Voisin and the first author in 2007.

Original language | English (US) |
---|---|

Pages (from-to) | 1819-1842 |

Number of pages | 24 |

Journal | Compositio Mathematica |

Volume | 147 |

Issue number | 6 |

DOIs | |

State | Published - Nov 1 2011 |

### Keywords

- p-adic neighbourhood
- rational maps
- rational points

### ASJC Scopus subject areas

- Algebra and Number Theory

## Fingerprint Dive into the research topics of 'Remarks on endomorphisms and rational points'. Together they form a unique fingerprint.

## Cite this

Amerik, E., Bogomolov, F., & Rovinsky, M. (2011). Remarks on endomorphisms and rational points.

*Compositio Mathematica*,*147*(6), 1819-1842. https://doi.org/10.1112/S0010437X11005537