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The classification of rational preperiodic points of quadratic polynomials over $$\mathbb{Q}$$: A refined conjecture. (English) Zbl 0902.11025
The author classifies the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $$\mathbb{Q}$$, assuming the conjecture that it is impossible to have rational points of period 4 or higher. In particular, he shows under this assumption that the number of preperiodic points is at most 9. Elliptic curves of small conductor and the genus 2 modular curves $$X_1(13)$$, $$X_1(16)$$, and $$X_1(18)$$ all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, he computes the rational points on a non-modular genus 2 curve by performing a 2-descent on its Jacobian and afterwards applying a variant of the method of Chabauty and Coleman.

##### MSC:
 11G30 Curves of arbitrary genus or genus $$\ne 1$$ over global fields 37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics 11G10 Abelian varieties of dimension $$> 1$$ 14H40 Jacobians, Prym varieties
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