Generators for the elliptic curve $y^2=x^3-nx$

Yasutsugu Fujita; Nobuhiro Terai

doi:10.5802/jtnb.769

Generators for the elliptic curve

y^{2} = x^{3} - n x

Yasutsugu Fujita ; Nobuhiro Terai

Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 403-416.

Résumé
Abstract

Soit $E$ la courbe elliptique définie par $y^{2} = x^{3} - n x$ où $n$ est un entier strictement positif. En $2007$ , Duquesne a démontré que, pour $k$ entier, si $n = (2 k^{2} - 2 k + 1) (18 k^{2} + 30 k + 17)$ est sans facteur carré, alors deux points rationnels spécifiques peuvent toujours se compléter en un système de générateurs du groupe de Mordell-Weil associé à $E$ . Dans ce papier, nous généralisons ce résultat en le montrant pour des entiers $n = n (k, l)$ pour une infinité de formes binaires $n (k, l) \in ℤ [k, l]$ .

Let $E$ be an elliptic curve given by $y^{2} = x^{3} - n x$ with a positive integer $n$ . Duquesne in 2007 showed that if $n = (2 k^{2} - 2 k + 1) (18 k^{2} + 30 k + 17)$ is square-free with an integer $k$ , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of $E$ . In this paper, we generalize this result and show that the same is true for infinitely many binary forms $n = n (k, l)$ in $ℤ [k, l]$ .

Reçu le : 2010-05-10
Publié le : 2017-03-02

MR 2817937 | Zbl 1228.11081

DOI : https://doi.org/10.5802/jtnb.769

@article{JTNB_2011__23_2_403_0,
     author = {Yasutsugu Fujita and Nobuhiro Terai},
     title = {Generators for the elliptic curve $y^2=x^3-nx$},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {403--416},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {23},
     number = {2},
     year = {2011},
     doi = {10.5802/jtnb.769},
     mrnumber = {2817937},
     zbl = {1228.11081},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/}
}

Yasutsugu Fujita; Nobuhiro Terai. Generators for the elliptic curve $y^2=x^3-nx$. Journal de Théorie des Nombres de Bordeaux, Tome 23 (2011) no. 2, pp. 403-416. doi : 10.5802/jtnb.769. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/

Bibliographie

[1] B. J. Birch and N. M. Stephens, The parity of the rank of the Mordell-Weil group. Topology 5 (1966), 296–299. | MR 201379 | Zbl 0146.42401

[2] W. Bosma and J. Cannon, Handbook of magma functions. Department of Mathematics, University of Sydney, available online at http://magma.maths.usyd.edu.au/magma/.

[3] J. W. S. Cassels, Introduction to the geometry of numbers. Springer-Verlag, 1959. | Zbl 0209.34401

[4] H. Cohen, A Course in computational algebraic number theory. Springer-Verlag, 1993. | MR 1228206 | Zbl 0786.11071

[5] S. Duquesne, Elliptic curves associated with simplest quartic fields. J. Theor. Nombres Bordeaux 19:1 (2007), 81–100. | Numdam | MR 2332055 | Zbl 1123.11018

[6] Y. Fujita and N. Terai, Integer points and independent points on the elliptic curve $y^{2} = x^{3} - p^{k} x$ . To appear in Tokyo J. Math. | MR 2731809

[7] F. Gouvêa and B. Mazur, The square-free sieve and the rank of elliptic curves. J. Amer. Math. Soc. 4:1 (1991), 1–23. | MR 1080648 | Zbl 0725.11027

[8] A. Granville, $A B C$ allows us to count squarefrees. Internat. Math. Res. Notices 19 (1998), 991–1009. | MR 1654759 | Zbl 0924.11018

[9] A. W. Knapp, Elliptic Curves. Princeton, Princeton Univ. Press, 1992. | MR 1193029 | Zbl 0804.14013

[10] M. Krir, À propos de la conjecture de Lang sur la minoration de la hauteur de Néron-Tate pour les courbes elliptiques sur $ℚ$ . Acta Arith. 100 (2001), 1–16. | MR 1864622 | Zbl 0981.11021

[11] S. Siksek, Infinite descent on elliptic curves. Rocky Mountain J. Math. 25:4 (1995), 1501–1538. | MR 1371352 | Zbl 0852.11028

[12] J. H. Silverman, The arithmetic of elliptic curves. Springer-Verlag, 1986. | MR 817210 | Zbl 1194.11005

[13] J. H. Silverman, Computing heights on elliptic curves. Math. Comp. 51 (1988), 339–358. | MR 942161 | Zbl 0656.14016

[14] J. H. Silverman, The advanced topics in the arithmetic of elliptic curves. Springer-Verlag, 1994. | MR 1312368 | Zbl 0911.14015

[15] J. Tate, Algorithm for determining the type of a singular fiber in an elliptic pencil. In Modular Functions of One Variable IV, Lecture Notes in Math. 476, Springer-Verlag, 1975, 33–52. | MR 393039