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

Abstract

For a quadratic endomorphism of the affine line defined over the rationals, we consider the problem of bounding the number of rational points that eventually land at the origin after iteration. In the article ``Uniform Bounds on Pre-Images Under Quadratic Dynamical Systems,'' by two of the present authors and five others, it was shown that the number of rational iterated pre-images of the origin is bounded as one varies the morphism in a certain one-dimensional family. Subject to the validity of the Birch and Swinnerton-Dyer conjecture and some other related conjectures for the L-series of a specific abelian variety and using a number of modern tools for locating rational points on high genus curves, we show that the maximum number of rational iterated pre-images is six. We also provide further insight into the geometry of the ``pre-image curves.''

Added M. Stoll as a coauthor, all open question in the paper have been resolved either unconditionally or else subject to standard conjectures for abelian varieties, Magma script for all computations included as an ancillary file