Uniformity in Mordell-Lang for curves

• 成果类型: Article
• 署名作者: Dimitrov, Vesselin; Gao, Ziyang; Habegger, Philipp
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2021.194.1.4
• 发表日期: 2021
• 页码: 237-298
• 关键词: rational-points; abelian-varieties; bounds; number; height; families

摘要

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 MordellWeil 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 AbelJacobi map. Both estimates generalize our previous work for one-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.