A consequence of the relative Bogomolov conjecture

TL;DR

This paper formulates a relative version of the Bogomolov conjecture and demonstrates its implications for the uniformity of the Mordell-Lang and Manin-Mumford conjectures for curves, advancing understanding in Diophantine geometry.

Contribution

It introduces a new formulation of the relative Bogomolov conjecture and links it to longstanding conjectures, providing a pathway to their resolution.

Findings

Relative Bogomolov conjecture implies uniform Manin-Mumford for curves

Establishes a connection between conjectures in Diophantine geometry

Builds on previous work to support the conjecture's implications

Abstract

We propose a formulation of the relative Bogomolov conjecture and show that it gives an affirmative answer to a question of Mazur's concerning the uniformity of the Mordell-Lang conjecture for curves. In particular we show that the relative Bogomolov conjecture implies the uniform Manin-Mumford conjecture for curves. The proof is built up on our previous work "Uniformity in Mordell-Lang for curves".