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

Document Type : Research Paper

Authors

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

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.

Keywords

Main Subjects

References

V. Bouniakowski, Sur les diviseurs numeriques invariables des functions rationelles entieres, Mem. Acad. Sci. St. Petersburg 6 (1857) 305--309.
H. Cohen, A Course in Computational Algebraic Number Theory, Springer-Verlag, Berlin, 1993.
J. Cannon, MAGMA Computational Algebra System, http://magma.maths.usyd.edu.au/magma/handbook/.
H. Daghigh, and S. Didari, On the elliptic curves of the form y2 = x3 -3px , Bull. Iranian Math. Soc. 40 (2014), no. 5, 1119--1133.
S. Duquesne, Elliptic curves associated with simplest quartic fields, J. Theor. Nombres Bordeaux 19 (2007), no. 1, 81--100.
Y. Fujita and T. Nara, On the Mordell-Weil group of the elliptic curve y2 = x3 + n, J. Number Theory 132 (2012), no. 3, 448--466.
Y. Fujita, N. Terai, Generators for the elliptic curve y2 = x3 -nx, J. Theor. Nombres Bordeaux 23 (2011), no. 2, 403--416.
Y. Fujita, Generators for the elliptic curve y2 = x3-nx of rank at least three, J. Number Theory 133 (2013), no. 5, 1645--1662.
C. Hooley, On the power-free values of polynomials in two variables, Analytic number theory, 235--266, Cambridge University Press, Cambridge, 2009.
J. Hoffstein, J. Pipher and J. H. Silverman, An Introduction to Mathematical Cryptography (Undergraduate Texts in Mathematics), Springer, New York, 2008.
A. W. Knapp, Elliptic Curves, Princeton University Press, Princeton, 1992.
T. Nagell, Zur Arithmetik der polynome, Abhandl. Math. Sem. Hamburg 1 (1922) 179--194.
S. Siksek, Infinite descent on elliptic curves, Rocky Mountain J. Math. 25 (1995), no. 4, 1501--1538.
J. H. Silverman, The Arithmetic of Elliptic Curves, Graduate Texts in Mathematics, 106, Springer, Dordrecht, 2009.
J. H. Silverman and J. Tate, Rational Points on Elliptic Curves, Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1992.
J. H. Silverman, Advanced topics in the arithmetic of elliptic curves, 151, Springer-Verlag, New York, 1994.
J.H. Silverman, Computing heights on elliptic curves, Math. Comp. 51 (1988), no. 183, 339--358.
L. C. Washington, Elliptic curves: Number Theory and Cryptography, Chapman & Hall/CRC, Boca Raton, 2008.

Files

History

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

Share

How to cite

Statistics