Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve
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 t...
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| Format: | Journal article |
| Language: | English |
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1995
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| _version_ | 1797071723288330240 |
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| author | Flynn, E Poonen, B Schaefer, E |
| author_facet | Flynn, E Poonen, B Schaefer, E |
| author_sort | Flynn, E |
| collection | OXFORD |
| 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 $N=6$. |
| first_indexed | 2024-03-06T22:57:26Z |
| format | Journal article |
| id | oxford-uuid:60ead365-6d94-4829-90a5-c316534d2742 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:57:26Z |
| publishDate | 1995 |
| record_format | dspace |
| spelling | oxford-uuid:60ead365-6d94-4829-90a5-c316534d27422022-03-26T17:56:13ZCycles of Quadratic Polynomials and Rational Points on a Genus-Two CurveJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:60ead365-6d94-4829-90a5-c316534d2742EnglishSymplectic Elements at Oxford1995Flynn, EPoonen, BSchaefer, EIt 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 $N=6$. |
| spellingShingle | Flynn, E Poonen, B Schaefer, E Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve |
| title | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve |
| title_full | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve |
| title_fullStr | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve |
| title_full_unstemmed | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve |
| title_short | Cycles of Quadratic Polynomials and Rational Points on a Genus-Two Curve |
| title_sort | cycles of quadratic polynomials and rational points on a genus two curve |
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