On the Number of Rational Iterated Pre-images of the Origin Under\n Quadratic Dynamical Systems

2008/10/09 by Xander Faber, Faber, Xander, Benjamin Hutz +3 · 1 citation
Mathematics · #11G18 #14G05 #37P99 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.0810.1715

openalex publication_date 2008/10/09 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

For a quadratic endomorphism of the affine line defined over the rationals,\nwe consider the problem of bounding the number of rational points that\neventually land at the origin after iteration. In the article ``Uniform Bounds\non Pre-Images Under Quadratic Dynamical Systems,'' by two of the present\nauthors and five others, it was shown that the number of rational iterated\npre-images of the origin is bounded as one varies the morphism in a certain\none-dimensional family. Subject to the validity of the Birch and\nSwinnerton-Dyer conjecture and some other related conjectures for the L-series\nof a specific abelian variety and using a number of modern tools for locating\nrational points on high genus curves, we show that the maximum number of\nrational iterated pre-images is six. We also provide further insight into the\ngeometry of the ``pre-image curves.''\n

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