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On the number of rational iterated preimages of the origin under quadratic dynamical systems. (English) Zbl 1242.14019
Given \(c \in \mathbb Q\) and a positive integer \(N\), let \(f_c\) be the endomorphism of the affine line \(\mathbb A^1_{\mathbb Q}\) defined by \(f_c (x) = x^2 +c\), and let \(f^N_c\) be the \(N\)-fold composition of \(f_c\). If \(f^{-N}_c\) denotes the \(N\)-fold preimage, the set of rational preimage of \(a \in \mathbb A^1 (\mathbb Q)\) is given by \[ \bigcup_{N \geq 1} f^{-N}_c (a) (\mathbb Q) = \{ x_0 \in \mathbb A^1 (\mathbb Q) \mid f^N_c (x_0) = a \; \text{ for some } \; a \in \mathbb A^1 (\mathbb Q) \} . \] This paper is concerned with the problem of bounding the number of rational points that eventually landing at the origin after iteration. Subject to the validity of the Birch-Swinnerton-Dyer conjecture and some other related conjectures for the \(L\)-series of a special abelian variety and using a number of modern tools for locating rational points on higher genus curves, the authors prove that the maximum number of rational iterated preimages is six. They also provide further insight into the geometry of the preimage curves.

14G05 Rational points
11G18 Arithmetic aspects of modular and Shimura varieties
11Y50 Computer solution of Diophantine equations
37F10 Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
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