## Title data

Poonen, Bjorn ; Stoll, Michael:

**Most odd degree hyperelliptic curves have only one rational point.**

*In:* Annals of Mathematics.
Vol. 180
(2014)
Issue 3
.
- pp. 1137-1166.

ISSN 0003-486X

DOI: https://doi.org/10.4007/annals.2014.180.3.7

## Project information

Project title: |
Project's official title Project's id Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory SPP 1489 |
---|---|

Project financing: |
Deutsche Forschungsgemeinschaft |

## Abstract in another language

Consider the smooth projective models C of curves y² = f(x) with f(x) ∈ ℤ[x] monic and separable of degree 2g + 1. We prove that for g ≥ 3, a positive fraction of these have only one rational point, the point at infinity. We prove a lower bound on this fraction that tends to 1 as g → ∞. Finally, we show that C(ℚ) can be algorithmically computed for such a fraction of the curves. The method can be summarized as follows: using p-adic analysis and an idea of McCallum, we develop a reformulation of Chabauty’s method that shows that certain computable conditions imply #C(ℚ) = 1; on the other hand, using further p-adic analysis, the theory of arithmetic surfaces, a new result on torsion points on hyperelliptic curves, and crucially the Bhargava-Gross theorems on the average number and equidistribution of nonzero 2-Selmer group elements, we prove that these conditions are often satisfied for p = 2.

## Further data

Item Type: | Article in a journal |
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Refereed: | Yes |

Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics II (Computer Algebra) > Chair Mathematics II (Computer Algebra) - Univ.-Prof. Dr. Michael Stoll Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics |

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 20 Nov 2014 10:44 |

Last Modified: | 20 Nov 2014 10:44 |

URI: | https://eref.uni-bayreuth.de/id/eprint/3748 |