Points rationnels et méthode de Chabauty elliptique

Duquesne, Sylvain

Journal de théorie des nombres de Bordeaux, Tome 15 (2003) no. 1, pp. 99-113.

La méthode de Chabauty elliptique permet de calculer les points rationnels sur une courbe définie sur un corps de nombres lorsque le théorème de Chabauty ne s’applique pas, c’est à dire lorsque le rang de la jacobienne est supérieur au genre de la courbe. Nous exposons cette méthode et nous la généralisons dans de nouveaux cas en écrivant une version explicite du théorème de préparation de Weierstrass en $2$ variables. En particulier nous calculons tous les points rationnels d’une courbe de genre $4$ dont le rang de la jacobienne vaut $4$.

The elliptic curve Chabauty method allows to compute rational points on curves defined over a number field when the rank of the jacobian is greater than the genus of the curve. We explain this method and generalize it to some new cases. In particular, we are able to compute rational points on a curve of genus $4$ and rank $4$.

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