TY - JOUR

T1 - Uniformity in Mordell-Lang for curves

AU - Dimitrov, Vesselin

AU - Gao, Ziyang

AU - Habegger, Philipp

N1 - Funding information:. The authors would like to thank Shou-wu Zhang for relevant discussions and Gabriel Dill for the argument involving torsion points to bound h1 on page 275. We would also like to thank Lars Kühne and Ngaim-ing Mok for discussions on Hermitian Geometry; our paper is much influenced by Kühne’s approach towards bounded height [Küh20] and by Mok’s approach to study the Mordell–Weil rank over function fields [Mok91]. We thank Gabriel Dill, Lars Kühne, Fabien Pazuki, and Joseph H. Silverman for corrections and comments on a draft of this paper. We also thank the referees for their careful reading and valuable comments. VD would like to thank the NSF and the Giorgio and Elena Petronio Fellowship Fund II for financial support for this work. VD has received funding from the European Union’s Seventh Frame-work Programme (FP7/2007–2013) / ERC grant agreement n? 617129. ZG has received funding from the French National Research Agency grant ANR-19-ERC7-0004, and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement n? 945714). PH has received funding from the Swiss National Science Foundation project n? 200020 184623. Both VD and ZG would like to thank the Institute for Advanced Study and the special year “Locally Symmetric Spaces: Analytical and Topological Aspects” for its hospitality during this work.

PY - 2021/7

Y1 - 2021/7

N2 - 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.

AB - 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.

KW - math.NT

KW - math.AG

KW - 11G30, 11G50, 14G05, 14G25

KW - Rational points

KW - Height inequality

KW - Mordell-lang

KW - Uniformity

UR - http://www.scopus.com/inward/record.url?scp=85129874988&partnerID=8YFLogxK

U2 - 10.4007/annals.2021.194.1.4

DO - 10.4007/annals.2021.194.1.4

M3 - Article

VL - 194

SP - 237

EP - 298

JO - Annals of Mathematics

JF - Annals of Mathematics

SN - 0003-486X

IS - 1

ER -