The Uniform Mordell-Lang Conjecture

Abstract

The Mordell--Lang conjecture for abelian varieties states that the intersection of an algebraic subvariety $X$ with a subgroup of finite rank is contained in a finite union of cosets contained in $X$. In this article, we prove a uniform version of this conjecture, meaning that that the number of cosets necessary does not depend on the ambient abelian variety. To achieve this, we prove a general gap principle on algebraic points that extends the gap principle for curves embedded into their Jacobians, previously obtained by Dimitrov--Gao--Habegger and Kühne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.

Accepted to Publications mathématiques de l'IHÉS. Comments are welcome!

References in corpus (5)

• Equidistribution in Families of Abelian Varieties and Uniformity
• Adelic line bundles on quasi-projective varieties
• Recent developments of the Uniform Mordell-Lang Conjecture
• Torsion points on isogenous abelian varieties
• Uniformity of quadratic points

Cited by in corpus (5)

• Equidistribution in Families of Abelian Varieties and Uniformity
• Ranks of abelian varieties in cyclotomic twist families
• Torsion points on isogenous abelian varieties
• Elliptic curves with common torsion $x$-coordinates and hyperelliptic torsion packets
• Uniform Mordell-Lang Plus Bogomolov