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Remarks on endomorphisms and rational points

Part of: Arithmetic and non-Archimedean dynamical systems Arithmetic problems. Diophantine geometry Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  24 August 2011

E. Amerik ,
F. Bogomolov  and
M. Rovinsky
E. Amerik
Affiliation:
Laboratoire de Mathématiques, Campus Scientifique d’Orsay, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France (email: Ekaterina.Amerik@math.u-psud.fr)
F. Bogomolov
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer Str., New York, NY 10012, USA Laboratory of Algebraic Geometry, GU-HSE, 7 Vavilova Str., Moscow 117312, Russia (email: bogomolo@cims.nyu.edu)
M. Rovinsky
Affiliation:
Independent University of Moscow, B. Vlasievsky Per. 11, 119002 Moscow, Russia Institute for Information Transmission Problems of Russian Academy of Sciences(email: marat@mccme.ru)
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Abstract

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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.

Keywords

rational pointsrational mapsp-adic neighbourhood

MSC classification

Secondary: 14G05: Rational points 11S82: Non-Archimedean dynamical systems 37P55: Arithmetic dynamics on general algebraic varieties
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 6 , November 2011 , pp. 1819 - 1842
Copyright
Copyright © Foundation Compositio Mathematica 2011

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