THE COMPLETE CLASSIFICATION OF RATIONAL PREPERIODIC POINTS OF QUADRATIC POLYNOMIALS OVER

Q: A REFINED CONJECTURE

BJORN POONEN

Abstract. 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 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, 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.