The Uniform Mordell-Lang Conjecture

TL;DR

This paper proves a uniform version of the Mordell-Lang conjecture for abelian varieties, establishing that the number of cosets needed is independent of the ambient variety, and also confirms the full uniform Bogomolov conjecture.

Contribution

It introduces a general gap principle on algebraic points that extends previous results and proves a uniform Mordell-Lang conjecture, also implying the uniform Bogomolov conjecture.

Findings

Proved a uniform Mordell-Lang conjecture for abelian varieties.

Established a new gap principle on algebraic points.

Confirmed the full uniform Bogomolov 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 K\"{u}hne. Our new gap principle also implies the full uniform Bogomolov conjecture in abelian varieties.