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

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