Uniformity in Mordell-Lang for curves

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

Authors

  • Vesselin Dimitrov
  • Ziyang Gao
  • Philipp Habegger

External Research Organisations

  • Centre national de la recherche scientifique (CNRS)
  • University of Basel
  • University of Toronto
  • Institut de mathématiques de Jussieu–Paris Rive Gauche (IMJ-PRG)
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Details

Original languageEnglish
Pages (from-to)237-298
Number of pages62
JournalAnnals of Mathematics
Volume194
Issue number1
Publication statusPublished - Jul 2021
Externally publishedYes

Abstract

Consider a smooth, geometrically irreducible, projective curve of genus \(g \ge 2\) defined over a number field of degree \(d \ge 1\). It has 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 \(g\), \(d\), and the Mordell-Weil rank of the curve's Jacobian, thereby answering in the affirmative a question of Mazur. In addition we obtain uniform bounded, in \(g\) and \(d\), for the number of geometric torsion points of the Jacobian which lie in the image of an Abel-Jacobi map. Both estimates generalize our previous work for \(1\)-parameter families. Our proof uses Vojta's approach to the Mordell Conjecture, and the key new ingredient is the generalization of a height inequality due to the second- and third-named authors.

Keywords

    math.NT, math.AG, 11G30, 11G50, 14G05, 14G25, Rational points, Height inequality, Mordell-lang, Uniformity

ASJC Scopus subject areas

Cite this

Uniformity in Mordell-Lang for curves. / Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp.
In: Annals of Mathematics, Vol. 194, No. 1, 07.2021, p. 237-298.

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

Dimitrov, V, Gao, Z & Habegger, P 2021, 'Uniformity in Mordell-Lang for curves', Annals of Mathematics, vol. 194, no. 1, pp. 237-298. https://doi.org/10.4007/annals.2021.194.1.4
Dimitrov, V., Gao, Z., & Habegger, P. (2021). Uniformity in Mordell-Lang for curves. Annals of Mathematics, 194(1), 237-298. https://doi.org/10.4007/annals.2021.194.1.4
Dimitrov V, Gao Z, Habegger P. Uniformity in Mordell-Lang for curves. Annals of Mathematics. 2021 Jul;194(1):237-298. doi: 10.4007/annals.2021.194.1.4
Dimitrov, Vesselin ; Gao, Ziyang ; Habegger, Philipp. / Uniformity in Mordell-Lang for curves. In: Annals of Mathematics. 2021 ; Vol. 194, No. 1. pp. 237-298.
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