# Uniform Manin-Mumford for a family of genus 2 curves

## TL;DR

This paper develops a strategy to establish uniform bounds on common points of height zero for pairs of height functions, applying it to prove conjectures related to torsion points and the Manin-Mumford conjecture for genus 2 curves.

## Contribution

It introduces a new general method for uniform bounds on height zero points and applies it to prove a conjecture for torsion points and uniform Manin-Mumford bounds for genus 2 curves.

## Key findings

- Proved a conjecture of Bogomolov, Fu, and Tschinkel on torsion points.
- Established uniform bounds for genus 2 curves over a7a7.
- Provided new quantitative bounds on common height zero points.

## Abstract

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogomolov bound for the family over $\overline{\mathbb{Q}}.$