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 $g\ge 2$, 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.

arXiv commentrevised

References in corpus (5)

• The Geometric Dynamical Northcott and Bogomolov Properties
• Adelic line bundles on quasi-projective varieties
• The Bounded Height Conjecture for Semiabelian Varieties
• The Uniform Mordell-Lang Conjecture
• Degeneration of Riemann theta functions and of the Zhang-Kawazumi invariant with applications to a uniform Bogomolov conjecture

Cited by in corpus (11)

• Adelic line bundles on quasi-projective varieties
• Recent developments of the Uniform Mordell-Lang Conjecture
• The Uniform Mordell-Lang Conjecture
• Ranks of abelian varieties in cyclotomic twist families
• Elliptic curves with common torsion $x$-coordinates and hyperelliptic torsion packets
• Torsion points on isogenous abelian varieties
• Arithmetic bigness and a uniform Bogomolov-type result
• Uniformity of quadratic points
• The tropical Manin-Mumford conjecture
• Intersecting the torsion of elliptic curves
• A uniform quantitative Manin-Mumford theorem for curves over function fields