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The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture
Bjorn Poonen
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We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over Q, assuming the conjecture that it is impossible to have rational points of period $4$ or higher. In particular, we show under this assumption that the number of preperiodic points is at most~$9$. Elliptic curves of small conductor and the genus~$2$ modular curves X₁(13), X₁(16), and X₁(18) all arise as curves classifying quadratic polynomials with various combinations of preperiodic points. To complete the classification, we compute 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.
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