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

TL;DR

This paper establishes an upper bound of six on the number of rational pre-images of the origin under quadratic dynamical systems over the rationals, assuming certain conjectures, and explores the geometry of related curves.

Contribution

It proves a conditional uniform bound of six rational pre-images for quadratic systems, advancing understanding of rational points in dynamical systems.

Findings

Maximum of six rational pre-images under certain conjectural assumptions.

Conditional proof based on Birch and Swinnerton-Dyer conjecture.

Insights into the geometry of pre-image curves.

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.''