Remarks on endomorphisms and rational points

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).

LaTeX, 22 pages. v2: some minor observations added, misprints corrected, appendix modified.

References in corpus (3)

• Questions on self maps of algebraic varieties
• Self Rational Maps of K3 Surfaces
• Flops on holomorphic symplectic fourfolds and determinantal cubic hypersurfaces

Cited by in corpus (11)

• Linear relations between polynomial orbits
• The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane
• Growth of the number of periodic points for meromorphic maps
• The existence of Zariski dense orbits for endomorphisms of projective surfaces (with an appendix in collaboration with Thomas Tucker)
• On the Medvedev-Scanlon Conjecture for Minimal Threefolds of Non-Negative Kodaira Dimension
• Examples of dynamical degree equals arithmetic degree
• Density of orbits of endomorphisms of abelian varieties
• Space spanned by characteristic exponents
• Potential density of projective varieties having an int-amplified endomorphism
• Algebraic curves $A^{\circ l}(x)-U(y)=0$ and arithmetic of orbits of rational functions
• Some applications of p-adic uniformization to algebraic dynamics