Equidistribution in Families of Abelian Varieties and Uniformity

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.