Daghigh, H., Didari, S. (2016). Complete characterization of the Mordell-Weil group of some families of elliptic curves. Bulletin of the Iranian Mathematical Society, 42(3), 585-594.

H. Daghigh; S. Didari. "Complete characterization of the Mordell-Weil group of some families of elliptic curves". Bulletin of the Iranian Mathematical Society, 42, 3, 2016, 585-594.

Daghigh, H., Didari, S. (2016). 'Complete characterization of the Mordell-Weil group of some families of elliptic curves', Bulletin of the Iranian Mathematical Society, 42(3), pp. 585-594.

Daghigh, H., Didari, S. Complete characterization of the Mordell-Weil group of some families of elliptic curves. Bulletin of the Iranian Mathematical Society, 2016; 42(3): 585-594.

Complete characterization of the Mordell-Weil group of some families of elliptic curves

^{}Faculty of Mathematical Sciences, University of Kashan, P.O. Box 8731751167, Kashan, Iran.

Receive Date: 10 February 2014,
Revise Date: 03 March 2015,
Accept Date: 07 March 2015

Abstract

The Mordell-Weil theorem states that the group of rational points on an elliptic curve over the rational numbers is a finitely generated abelian group. In our previous paper, H. Daghigh, and S. Didari, On the elliptic curves of the form $ y^2=x^3-3px$, Bull. Iranian Math. Soc. 40 (2014), no. 5, 1119--1133., using Selmer groups, we have shown that for a prime $p$ the rank of elliptic curve $y^2=x^3-3px$ is at most two. In this paper we go further, and using height function, we will determine the Mordell-Weil group of a family of elliptic curves of the form $y^2=x^3-3nx$, and give a set of its generators under certain conditions. We will introduce an infinite family of elliptic curves with rank at least two. The full Mordell-Weil group and the generators of a family (which is expected to be infinite under the assumption of a standard conjecture) of elliptic curves with exact rank two will be described.

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