Flynn, E. V. and Poonen, B. and Schaefer, E. F. (1997) *Cycles of quadratic polynomials and rational points on a genus 2 curve.* Duke Mathematical Journal, 90 . pp. 435-463. ISSN 0012-7094

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## 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(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.

Item Type: | Article |
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Uncontrolled Keywords: | Arithmetic dynamics; periodic point; descent; hyperelliptic curve; method of Chabauty and Coleman;uniform boundedness; modular curve. |

Subjects: | A - C > Algebraic geometry H - N > Number theory D - G > Dynamical systems and ergodic theory |

Research Groups: | Number Theory Group |

ID Code: | 263 |

Deposited By: | E. Victor Flynn |

Deposited On: | 12 Jul 2006 |

Last Modified: | 20 Jul 2009 14:19 |

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