Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve
| dc.creator | Flynn, E. V. | |
| dc.creator | Poonen, Bjorn | |
| dc.creator | Schaefer, Edward F. | |
| dc.date | 1995-08-04 | |
| dc.date.accessioned | 2026-07-07T09:15:23Z | |
| dc.date.available | 2026-07-07T09:15:23Z | |
| dc.description | 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 $X_1(16)$, 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$_{\bold Q}$-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 $N=6$. | |
| dc.identifier | https://arxiv.org/abs/math/9508211 | |
| dc.identifier | http://arxiv.org/abs/math/9508211 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/152999 | |
| dc.subject | Number Theory | |
| dc.title | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve | |
| dc.type | text |