1995/08/04 by E. V. Flynn, Bjorn Poonen, Flynn, E. V. +3 · 5 citations
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.math/9508211
openalex publication_date 1995/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It has been conjectured that for N sufficiently large, there are no quadratic polynomials in \bold 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 X1(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\bold Q-stable 5-cycles, and show that there exist Gal\bold 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.