Elliptic curves associated with simplest quartic fields

Duquesne, Sylvain

Journal de théorie des nombres de Bordeaux, Tome 19 (2007) no. 1, pp. 81-100.

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.

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.

[1] H. Cohen, A Course in Computational Algebraic Number Theory. Graduate Texts in Math. 138, Springer-Verlag, 1993. | MR | Zbl

[2] J. Cremona, M. Prickett, S. Siksek, Height difference bounds for elliptic curves over number fields. Journal of Number Theory 116 (2006), 42–68. | MR | Zbl

[3] J. Cremona, S. Siksek, Computing a Lower Bound for the Canonical Height on Elliptic Curves over $\mathbb{Q}$. Algorithmic Number Theory, 7th International Symposium, ANTS-VII, LNCS 4076 (2006), 275–286. | MR

[4] S. Duquesne, Integral points on elliptic curves defined by simplest cubic fields. Exp. Math. 10:1 (2001), 91–102. | MR | Zbl

[5] M. N. Gras, Table numérique du nombre de classes et des unités des extensions cycliques réelles de degré 4 de $\mathbb{Q}$. Publ. Math. Fac. Sci. Besancon, fasc 2 (1977/1978). | MR | Zbl

[6] E. Halberstadt, Signes locaux des courbes elliptiques en 2 et 3. C. R. Acad. Sci. Paris Sér. I Math. 326:9 (1998), 1047–1052. | MR | Zbl

[7] H. K. Kim, Evaluation of zeta functions at $s=-1$ of the simplest quartic fields. Proceedings of the 2003 Nagoya Conference “Yokoi-Chowla Conjecture and Related Problems”, Saga Univ., Saga, 2004, 63–73.

[8] A. J. Lazarus, Class numbers of simplest quartic fields. Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, 313–323. | MR | Zbl

[9] A. J. Lazarus, On the class number and unit index of simplest quartic fields. Nagoya Math. J. 121 (1991), 1–13. | MR | Zbl

[10] S. Louboutin, The simplest quartic fields with ideal class groups of exponents less than or equal to 2. J. Math. Soc. Japan 56:3 (2004), 717–727. | MR | Zbl

[11] P. Olajos, Power integral bases in the family of simplest quartic fields. Experiment. Math. 14:2 (2005), 129–132. | MR | Zbl

[12] O. Rizzo, Average root numbers for a nonconstant family of elliptic curves. Compositio Math. 136:1 (2003), 1–23. | MR | Zbl

[13] S. Siksek, Infinite descent on elliptic curves. Rocky Mountain J. Math. 25:4 (1995), 1501–1538. | MR | Zbl

[14] J. H. Silverman, The arithmetic of elliptic curves. Graduate Texts in Mathematics 106, Springer-Verlag, 1986. | MR | Zbl

[15] J. H. Silverman, Computing heights on elliptic curves. Math. Comp. 51 (1988), 339–358. | MR | Zbl

[16] J. H. Silverman, The difference between the Weil height and the canonical height on elliptic curves. Math. Comp. 55 (1990), 723–743. | MR | Zbl