Remarks on endomorphisms and rational points

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.

The existence of Zariski dense orbits for polynomial endomorphisms of the affine plane

“…This conjecture was proposed by Medvedev and Scanlon [25,Conjecture 5.10] and also by Amerik, Bogomolov and Rovinsky [2], which strengthens the following conjecture of Zhang [33].…”

“…If both |λ 1 | and |λ 2 | are strictly less than 1, since λ 1 , λ 2 are not multiplicatively independent, there exists m 1 , m 2 ≥ 1 such that λ m 1 1 = λ m 2 2 . after replacing f by a suitable iterate, we may assume that (m 1 , m 2 ) = 1.…”

Growth of the number of periodic points for meromorphic maps

“…We expect that our techniques may be extended to study other dynamical questions of algebraic nature. The readers can find in a list of open problems, related results and references.…”

A case of the dynamical Mordell–Lang conjecture

“…Hence assume ω n ≥ g n for all large n. Write u n = β n +t ω n ρ n . Note that ρ n does not vanish at 0 and has bounded degree because a n = deg u n and ω n ≥ g n ≥ a n − O (1). Now, u n+1 = ϕ(u n ), hence…”

“…Also, in [36], Zhang formulates a conjecture about the Zariski density of orbits of points under fairly general maps from a projective variety to itself. Amerik, Bogomolov, and Rovinsky [1,2] have obtained partial results towards this conjecture, using p-adic methods similar to those used in this paper. This latter conjecture of Zhang takes the following form in the case of coordinatewise polynomial actions on A g .…”