Uniformity in Mordell-Lang for curves

Vesselin Dimitrov; Ziyang Gao; Philipp Habegger

doi:10.4007/annals.2021.194.1.4

Details

Original language	English
Pages (from-to)	237-298
Number of pages	62
Journal	Annals of Mathematics
Volume	194
Issue number	1
Publication status	Published - Jul 2021
Externally published	Yes

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

Mathematics(all)
Statistics and Probability
Decision Sciences(all)
Statistics, Probability and Uncertainty

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.

Download

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$g \ge 2$ 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

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Research@Leibniz University

Uniformity in Mordell-Lang for curves

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ASJC Scopus subject areas

Cite this