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Uniform Bound for the Number of Rational Points on a Pencil of Curves

Dimitrov, Vesselin and Gao, Ziyang and Habegger, Philipp. (2019) Uniform Bound for the Number of Rational Points on a Pencil of Curves. International mathematics research notices. rnz248.

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Official URL: https://edoc.unibas.ch/75076/

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Abstract

Consider a one-parameter family of smooth, irreducible, projective curves of genus g≥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 2nd- and 3rd-named authors. In addition we obtain uniform bounds for the number of torsion points in the Jacobian that lie in each fiber of the family.
Faculties and Departments:05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Zahlentheorie (Habegger)
UniBasel Contributors:Habegger, Philipp
Item Type:Article, refereed
Article Subtype:Research Article
Publisher:Hindawi Publ. Corp.
ISSN:1073-7928
Note:Publication type according to Uni Basel Research Database: Journal article
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Last Modified:25 Jun 2020 13:11
Deposited On:24 Jun 2020 15:46

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