La conjecture de Mordell: origines, approches, g\'en\'eralisations

TL;DR

The paper reviews the history, approaches, and generalizations of the Mordell conjecture, highlighting its significance in number theory and the ongoing conjectural extensions developed over the past century.

Contribution

It provides a comprehensive overview of the origins, various proofs, and extensions of the Mordell conjecture, emphasizing the conjectural nature of many recent developments.

Findings

Multiple approaches and proofs have been developed since 1922.

Many extensions of the Mordell conjecture remain conjectural.

The conjecture has significantly influenced the development of modern number theory.

Abstract

The Mordell conjecture: origins, approaches, generalizations -- The Mordell conjecture predicts that a diophantine equation defining a smooth projective curve of genus at least two has only finity many solutions in a given number field. The century that ran since its statement, in 1922, gave rise to several approaches, several proofs, and vast extensions most of which are still conjectural. This text is based on the oral presentation and aims at recalling this story. La conjecture de Mordell pr\'edit qu'une \'equation diophantienne d\'efinissant une courbe projective lisse de genre au moins deux n'a qu'un nombre fini de solutions dans un corps de nombres donn\'e. Le si\`ecle qui s'est \'ecoul\'e depuis son \'enonc\'e, en 1922, a vu plusieurs approches, plusieurs d\'emonstrations, ainsi que de vastes extensions dont la plupart sont encore conjecturales. Ce texte, qui reprend l'expos\'e oral, s'efforce de retracer cette histoire.