Bornes sur le nombre de points rationnels des courbes -- en quete d'uniformite

TL;DR

This paper explores how a conjectural lower bound on the canonical height function can lead to an explicit uniform bound on the number of rational points on algebraic curves of genus at least 2 over number fields.

Contribution

It establishes a connection between a conjectural height bound and uniform bounds on rational points, advancing understanding in Diophantine geometry.

Findings

Proposes a link between height conjectures and rational point bounds

Derives explicit bounds assuming conjectural height lower bounds

Contributes to the uniformity problem in Diophantine geometry

Abstract

The aim of this paper is to show how a conjectural lower bound on the canonical height function in the spirit of Lang and Silverman leads to an explicit uniform bound on the number of rational points on curves of genus $g\geq 2$ over a number field.