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arXiv · 2101.10272

Equidistribution in Families of Abelian Varieties and Uniformity

Lars Kühne
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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 $g \geq 2$, 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.

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Lars Kühne. 2021-01-25. Equidistribution in Families of Abelian Varieties and Uniformity. https://arxiv.org/abs/2101.10272

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