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. 435463. ISSN 00127094

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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 2descent 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 Galoisstable 5cycles, and show that there exist Galoisstable Ncycles 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:  29 May 2015 18:19 
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