Elliptic curves associated with simplest quartic fields
Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 81-100.

We are studying the infinite family of elliptic curves associated with simplest cubic fields. If the rank of such curves is 1, we determine the whole structure of the Mordell-Weil group and find all integral points on the original model of the curve. Note however, that we are not able to find them on the Weierstrass model if the parameter is even. We have also obtained similar results for an infinite subfamily of curves of rank 2. To our knowledge, this is the first time that so much information has been obtained both on the structure of the Mordell-Weil group and on integral points for an infinite family of curves of rank 2. The canonical height is the main tool we used for that study.

Nous étudions la famille infinie des courbes elliptiques associées aux “simplest quartic fields”. Si le rang de telles courbes vaut 1, nous déterminons la structure complète du groupe de Mordell-Weil et nous trouvons tous les points entiers sur le modèle original de la courbe. Notons toutefois que nous ne sommes pas capables de les trouver sur le modèle de Weierstrass quand le paramètre est pair. Nous obtenons également des résultats similaires pour une sous-famille infinie de courbes de rang 2. A notre connaissance, c’est la première fois que l’on a autant d’information sur la structure du groupe de Mordell-Weil et sur les points entiers pour une famille infinie de courbes de rang 2. Le principal outils que nous avons utilisé pour cette étude est la hauteur canonique.

Received: 2005-12-28
Published online: 2008-12-03
DOI: https://doi.org/10.5802/jtnb.575
@article{JTNB_2007__19_1_81_0,
     author = {Sylvain Duquesne},
     title = {Elliptic curves associated with simplest quartic fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {19},
     number = {1},
     year = {2007},
     pages = {81-100},
     doi = {10.5802/jtnb.575},
     zbl = {1123.11018},
     mrnumber = {2332055},
     language = {en},
     url={jtnb.centre-mersenne.org/item/JTNB_2007__19_1_81_0/}
}
Duquesne, Sylvain. Elliptic curves associated with simplest quartic fields. Journal de Théorie des Nombres de Bordeaux, Volume 19 (2007) no. 1, pp. 81-100. doi : 10.5802/jtnb.575. https://jtnb.centre-mersenne.org/item/JTNB_2007__19_1_81_0/

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