## Title data

Stoll, Michael:

**Uniform bounds for the number of rational points on hyperelliptic curves of small Mordell-Weil rank.**

*In:* Journal of the European Mathematical Society.
Vol. 21
(2019)
Issue 3
.
- pp. 923-956.

ISSN 1435-9855

DOI: https://doi.org/10.4171/JEMS/857

## Abstract in another language

We show that there is a bound depending only on g,r and [K:Q] for the number of K-rational points on a hyperelliptic curve C of genus g over a number field K such that the Mordell–Weil rank r of its Jacobian is at most g–3. If K=Q, an explicit bound is 8rg+33(g–1)+1.

The proof is based on Chabauty’s method; the new ingredient is an estimate for the number of zeros of an abelian logarithm on a p-adic ‘annulus’ on the curve, which generalizes the standard bound on disks. The key observation is that for a p-adic field k, the set of k-points on C can be covered by a collection of disks and annuli whose number is bounded in terms of g (and k).

We also show, strengthening a recent result by Poonen and the author, that the lower density of hyperelliptic curves of odd degree over Q whose only rational point is the point at infinity tends to 1 uniformly over families defined by congruence conditions, as the genus g tends to infinity.

## Further data

Item Type: | Article in a journal |
---|---|

Refereed: | Yes |

Keywords: | Rational points on curves; uniform bounds; Chabauty’s method; p-adic integration; Mordell–Lang conjecture; Zilber–Pink conjectures |

Subject classification: | Mathematics Subject Classification (2010): Primary 11G30, 14G05; Secondary 14G25, 14H25, 14H40 |

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

Result of work at the UBT: | Yes |

DDC Subjects: | 500 Science > 510 Mathematics |

Date Deposited: | 19 Aug 2019 08:03 |

Last Modified: | 19 Aug 2019 08:03 |

URI: | https://eref.uni-bayreuth.de/id/eprint/51924 |