Uniformity in Mordell-Lang for curves

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

Autoren

  • Vesselin Dimitrov
  • Ziyang Gao
  • Philipp Habegger

Externe Organisationen

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

OriginalspracheEnglisch
Seiten (von - bis)237-298
Seitenumfang62
FachzeitschriftAnnals of Mathematics
Jahrgang194
Ausgabenummer1
PublikationsstatusVeröffentlicht - Juli 2021
Extern publiziertJa

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.

ASJC Scopus Sachgebiete

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Uniformity in Mordell-Lang for curves. / Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp.
in: Annals of Mathematics, Jahrgang 194, Nr. 1, 07.2021, S. 237-298.

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

Dimitrov, V, Gao, Z & Habegger, P 2021, 'Uniformity in Mordell-Lang for curves', Annals of Mathematics, Jg. 194, Nr. 1, S. 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 ; Jahrgang 194, Nr. 1. S. 237-298.
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