Let be an elliptic curve given by with a positive integer . Duquesne in 2007 showed that if is square-free with an integer , then certain two rational points of infinite order can always be in a system of generators for the Mordell-Weil group of . In this paper, we generalize this result and show that the same is true for infinitely many binary forms in .
Soit la courbe elliptique définie par où est un entier strictement positif. En , Duquesne a démontré que, pour entier, si 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 . Dans ce papier, nous généralisons ce résultat en le montrant pour des entiers pour une infinité de formes binaires .
@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/} }
TY - JOUR TI - Generators for the elliptic curve $y^2=x^3-nx$ JO - Journal de Théorie des Nombres de Bordeaux PY - 2011 DA - 2011/// SP - 403 EP - 416 VL - 23 IS - 2 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/ UR - https://www.ams.org/mathscinet-getitem?mr=2817937 UR - https://zbmath.org/?q=an%3A1228.11081 UR - https://doi.org/10.5802/jtnb.769 DO - 10.5802/jtnb.769 LA - en ID - JTNB_2011__23_2_403_0 ER -
Yasutsugu Fujita; Nobuhiro Terai. Generators for the elliptic curve $y^2=x^3-nx$. Journal de Théorie des Nombres de Bordeaux, Volume 23 (2011) no. 2, pp. 403-416. doi : 10.5802/jtnb.769. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.769/
[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 . 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, 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 . 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
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