Uniform Mordell-Lang Plus Bogomolov

TL;DR

This paper establishes a uniform version of the Mordell-Lang plus Bogomolov theorem for abelian varieties, extending previous results to include an epsilon-neighborhood, with implications for the distribution of rational points.

Contribution

It generalizes Rémond's work on large points by incorporating an epsilon-neighborhood, providing a more comprehensive uniform theorem for abelian varieties.

Findings

Proves a uniform Mordell-Lang plus Bogomolov theorem for abelian varieties.

Extends Rémond's results to include epsilon-neighborhoods.

Builds on previous work on small points with Gao and Kühne.

Abstract

In this paper, we prove a uniform version of Poonen's "Mordell-Lang Plus Bogomolov" theorem for abelian varieties. We mainly generalize R\'emond's work on large points to allow an extra $\epsilon$-neighborhood. The part on small points follows from an earlier paper, joint with Gao and K\"uhne.