Minoration de la hauteur de Neron-Tate sur les surfaces abeliennes

TL;DR

This paper investigates a conjecture on lower bounds for the Neron-Tate height on abelian surfaces, using local height decomposition to derive bounds on torsion points and rational points.

Contribution

It introduces a new approach using local height decomposition to establish lower bounds for heights on abelian surfaces, supporting the Lang-Silverman conjecture.

Findings

Derived uniform bounds on torsion points for abelian surfaces

Established bounds on rational points on genus 2 curves

Provided evidence supporting the Lang-Silverman conjecture

Abstract

This paper contains results concerning a conjecture made by Lang and Silverman predicting a lower bound for the canonical height on abelian varieties of dimension 2 over number fields. The method used here is a local height decomposition. We derive as corollaries uniform bounds on the number of torsion points on families of abelian surfaces and on the number of rational points on families of genus 2 curves.