## Details

Originalsprache | Englisch |
---|---|

Seiten (von - bis) | 1138–1159 |

Seitenumfang | 22 |

Fachzeitschrift | International Mathematics Research Notices |

Jahrgang | 2021 |

Ausgabenummer | 2 |

Frühes Online-Datum | 10 Dez. 2019 |

Publikationsstatus | Veröffentlicht - Jan. 2021 |

Extern publiziert | Ja |

## Abstract

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**Uniform bound for the number of rational points on a pencil of curves.**/ Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp.

in: International Mathematics Research Notices, Jahrgang 2021, Nr. 2, 01.2021, S. 1138–1159.

Publikation: Beitrag in Fachzeitschrift › Artikel › Forschung › Peer-Review

*International Mathematics Research Notices*, Jg. 2021, Nr. 2, S. 1138–1159. https://doi.org/10.1093/imrn/rnz248

*International Mathematics Research Notices*,

*2021*(2), 1138–1159. https://doi.org/10.1093/imrn/rnz248

}

TY - JOUR

T1 - Uniform bound for the number of rational points on a pencil of curves

AU - Dimitrov, Vesselin

AU - Gao, Ziyang

AU - Habegger, Philipp

N1 - Publisher Copyright: © 2021 The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.

PY - 2021/1

Y1 - 2021/1

N2 - Consider a one-parameter family of smooth, irreducible, projective curves of genus \(g\ge 2\) defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber of the family.

AB - Consider a one-parameter family of smooth, irreducible, projective curves of genus \(g\ge 2\) defined over a number field. Each fiber contains at most finitely many rational points by the Mordell Conjecture, a theorem of Faltings. We show that the number of rational points is bounded only in terms of the family and the Mordell--Weil rank of the fiber's Jacobian. Our proof uses Vojta's approach to the Mordell Conjecture furnished with a height inequality due to the second- and third-named authors. In addition we obtain uniform bounds for the number of torsion in the Jacobian that lie each fiber of the family.

KW - math.NT

KW - math.AG

KW - 11G30, 11G50, 14G05, 14G25

UR - http://www.scopus.com/inward/record.url?scp=85108805672&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz248

DO - 10.1093/imrn/rnz248

M3 - Article

VL - 2021

SP - 1138

EP - 1159

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 2

ER -