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

Research output: Contribution to journal › Article › Research › peer review

Authors

  • Vesselin Dimitrov
  • Ziyang Gao
  • Philipp Habegger

External Research Organisations

  • University of Cambridge
  • Centre national de la recherche scientifique (CNRS)
  • University of Basel
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Details

Original languageEnglish
Pages (from-to)1138–1159
Number of pages22
JournalInternational Mathematics Research Notices
Volume2021
Issue number2
Early online date10 Dec 2019
Publication statusPublished - Jan 2021
Externally publishedYes

Abstract

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.

Keywords

    math.NT, math.AG, 11G30, 11G50, 14G05, 14G25

ASJC Scopus subject areas

Cite this

Uniform bound for the number of rational points on a pencil of curves. / Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp.
In: International Mathematics Research Notices, Vol. 2021, No. 2, 01.2021, p. 1138–1159.

Research output: Contribution to journal › Article › Research › peer review

Dimitrov V, Gao Z, Habegger P. Uniform bound for the number of rational points on a pencil of curves. International Mathematics Research Notices. 2021 Jan;2021(2):1138–1159. Epub 2019 Dec 10. doi: 10.1093/imrn/rnz248
Dimitrov, Vesselin ; Gao, Ziyang ; Habegger, Philipp. / Uniform bound for the number of rational points on a pencil of curves. In: International Mathematics Research Notices. 2021 ; Vol. 2021, No. 2. pp. 1138–1159.
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