Density of orbits of endomorphisms of abelian varieties

2014/12/05 by Dragos Ghioca, Thomas Scanlon, Ghioca, Dragos +1
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation #math.AG #math.NT

paper · pdf · doi:10.48550/arxiv.1412.2029

arxiv created 2014/12/05 · openalex publication_date 2014/12/05 · arxiv updated 2014/12/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

Let A be an abelian variety defined over ℚ, and let φ be a dominant endomorphism of A as an algebraic variety. We prove that either there exists a non-constant rational fibration preserved by φ, or there exists a point x∈ A(ℚ) whose φ-orbit is Zariski dense in A. This provides a positive answer for abelian varieties of a question raised by Medvedev and the second author ("nvariant varieties for polynomial dynamical systems", Ann. of Math. (2) 179 (2014), no. 1, 81-177). We prove also a stronger statement of this result in which φ is replaced by any commutative finitely generated monoid of dominant endomorphisms of A.

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