On the rank of certain parametrized elliptic curves

Subject Areas : Applied Mathematics

Ali Hadavand ¹ (aDepartment of mathematics, Arak Branch, Islamic Azad university, Arak, Iran.)

Received: 2016-08-24 Accepted : 2015-08-14 Published : 2016-12-01

Keywords: Elliptic Curve, Selmer Group,

Abstract :

In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.

References:

[1] J.S. Chahal, Topics in number theory, Kluwer Academic/Plenum
Publisher, 1988.
[2] J. E. Cremona, Algorithms of modular elliptic curves, Cambridge
University Press, 1997.
[3] A.J. Hollier, B. K. Spearman and Q. Yang, On the rank and integral points
of Elliptic Curves y² = x³ - px, Int. J. Algebra, 3 ( 2009), no. 8, 401-406.
[4] T. Kudo and K. Motose, On Group structures of some special elliptic
curves,Math. J. Okayama Univ. 47 (2005), 81-84.
[5] S. Schmitt, Computing of the Selmer groups of certain parametrized
elliptic curves, Acta Arithmetica. LXXVIII (1997) 241{254.
[6] J. H. Silverman, The Arithmetic of Elliptic curves, GTM 106. Springer-
Verlage, New York, 1986.
[7] J. H. Silverman and J. Tate, Rational Points on Elliptic curves, Springer-
Verlage, New York, 1994.
21
[8] R. J. Stroeker and J. Top, On the equation y² = (X +p)(X² +p²), Rocky
Mountain J. Math. 27 (1994), 1135{1161.
[9] P. G. Walsh, Maximal Ranks and Integer Points on a Family of Elliptic
Curves,Glasnik Mathematicki. 44 (2009), 83-87.