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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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Abstract

We classify the graphs that can occur as the graph of rational preperiodic points of a quadratic polynomial over $\bold Q$, 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.

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BibTeXRIS

Bjorn Poonen. 1995-12-11. The Complete Classification of Rational Preperiodic Points of Quadratic Polynomials over Q: A Refined Conjecture. https://arxiv.org/abs/math/9512217

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