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 = {Fujita, Yasutsugu and Terai, Nobuhiro}, 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}, zbl = {1228.11081}, mrnumber = {2817937}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.769/} }
TY - JOUR AU - Fujita, Yasutsugu AU - Terai, Nobuhiro TI - Generators for the elliptic curve $y^2=x^3-nx$ JO - Journal de théorie des nombres de Bordeaux PY - 2011 SP - 403 EP - 416 VL - 23 IS - 2 PB - Société Arithmétique de Bordeaux UR - http://archive.numdam.org/articles/10.5802/jtnb.769/ DO - 10.5802/jtnb.769 LA - en ID - JTNB_2011__23_2_403_0 ER -
%0 Journal Article %A Fujita, Yasutsugu %A Terai, Nobuhiro %T Generators for the elliptic curve $y^2=x^3-nx$ %J Journal de théorie des nombres de Bordeaux %D 2011 %P 403-416 %V 23 %N 2 %I Société Arithmétique de Bordeaux %U http://archive.numdam.org/articles/10.5802/jtnb.769/ %R 10.5802/jtnb.769 %G en %F JTNB_2011__23_2_403_0
Fujita, Yasutsugu; Terai, Nobuhiro. 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. http://archive.numdam.org/articles/10.5802/jtnb.769/
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