The Mathematical Institute, University of Oxford, Eprints Archive

Cycles of quadratic polynomials and rational points on a genus 2 curve

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$_1$(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.

Item Type:Article
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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