Uniformity in Mordell–Lang for curves - Sorbonne Université
Journal Articles Annals of Mathematics Year : 2021

Uniformity in Mordell–Lang for curves

Abstract

Consider a smooth, geometrically irreducible, projective curve of genus g ≥ 2 defined over a number field of degree d ≥ 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 bounds, 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.
Fichier principal
Vignette du fichier
Dimitrov et al. - 2021 - Uniformity in Mordell–Lang for curves.pdf (661.61 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03374335 , version 1 (12-10-2021)

Identifiers

Cite

Vesselin Dimitrov, Ziyang Gao, Philipp Habegger. Uniformity in Mordell–Lang for curves. Annals of Mathematics, 2021, 194 (1), pp.237-298. ⟨10.4007/annals.2021.194.1.4⟩. ⟨hal-03374335⟩
24 View
217 Download

Altmetric

Share

More