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 <firstname.lastname@example.org>
Bjorn Poonen <email@example.com>
Edward F. Schaefer <firstname.lastname@example.org>