Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve

Abstract

It has been conjectured that for $N$ sufficiently large, there are no quadratic polynomials in $\mathbf{Q}[z]$ 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${}_{\mathbf{Q}}$-stable $5$-cycles, and show that there exist Gal${}_{\mathbf{Q}}$-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 $N=6$.