Cycles of quadratic polynomials and rational points on a genus 2 curve, by E. V. Flynn, Bjorn Poonen, and Edward F. Schaefer

This paper has appeared in Duke Math. J. 98 (1997), no. 3, 435-463 and so the dvi version has been removed. 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 Galois-stable 5-cycles, and show that there exist Galois-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.

E. V. Flynn <>
Bjorn Poonen <>
Edward F. Schaefer <>