### 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 2011 |

### Fingerprint

### Keywords

- p-adic neighbourhood
- rational maps
- rational points

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

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

**Remarks on endomorphisms and rational points.** / Amerik, E.; Bogomolov, Fedor; Rovinsky, M.

Research output: Contribution to journal › Article

*Compositio Mathematica*, vol. 147, no. 6, pp. 1819-1842. https://doi.org/10.1112/S0010437X11005537

}

TY - JOUR

T1 - Remarks on endomorphisms and rational points

AU - Amerik, E.

AU - Bogomolov, Fedor

AU - Rovinsky, M.

PY - 2011/11

Y1 - 2011/11

N2 - 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.

AB - 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.

KW - p-adic neighbourhood

KW - rational maps

KW - rational points

UR - http://www.scopus.com/inward/record.url?scp=82955174756&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=82955174756&partnerID=8YFLogxK

U2 - 10.1112/S0010437X11005537

DO - 10.1112/S0010437X11005537

M3 - Article

AN - SCOPUS:82955174756

VL - 147

SP - 1819

EP - 1842

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 6

ER -