CYCLES OF QUADRATIC POLYNOMIALS AND

RATIONAL POINTS ON A GENUS-TWO CURVE

E. V. Flynn Bjorn Poonen Edward F. Schaefer

Abstract. It has been conjectured that for N sufficiently large, there are no quadratic polynomials in Q[z] with rational periodic points of period N. Morton proved there were none with N = 4, by showing that the genus 2 algebraic curve that classifies periodic points of period 4 is birational to X_{1}(16), whose rational points had been previously computed. We prove there are none with N = 5. Here the relevant curve has genus 14, but it has a genus 2 quotient, whose rational points we compute by performing a 2-descent on its Jacobian and applying a refinement of the method of Chabauty and Coleman. We hope that our computation will serve as a model for others who need to compute rational points on hyperelliptic curves. We also describe the three possible Gal_{Q}-stable 5-cycles, and show that there exist Gal_{Q}- stable N-cycles for infinitely many N. Furthermore, we answer a question of Morton by showing that the genus 14 curve and its quotient are not modular. Finally, we mention some partial results for N = 6.

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