Equidistribution in Families of Abelian Varieties and Uniformity

2021/01/25 by Lars Kühne, Kühne, Lars · 5 citations
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Coding theory and cryptography #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2101.10272

Abstract

Using equidistribution techniques from Arakelov theory as well as recent results obtained by Dimitrov, Gao, and Habegger, we deduce uniform results on the Manin-Mumford and the Bogomolov conjecture. For each given integer g ≥ 2, we prove that the number of torsion points lying on a smooth complex algebraic curve of genus g embedded into its Jacobian is uniformly bounded. Complementing recent works of Dimitrov, Gao, and Habegger, we obtain a rather uniform version of the Mordell conjecture as well. In particular, the number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian.

Citations

• Courbes sur une vari�t� ab�lienne et points de torsion

Cited by

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• Counting rational points on smooth quartic and quintic surfaces
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• The Uniform Mordell-Lang Conjecture
• On Vojta's proof of the Mordell conjecture

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