Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31903
The Elliptic Curves y2 = x3 - t2x over Fp

Authors: Ahmet Tekcan


Let p be a prime number, Fp be a finite field and t ∈ F*p= Fp- {0}. In this paper we obtain some properties of ellipticcurves Ep,t: y2= y2= x3- t2x over Fp. In the first sectionwe give some notations and preliminaries from elliptic curves. In the second section we consider the rational points (x, y) on Ep,t. Wegive a formula for the number of rational points on Ep,t over Fnp for an integer n ≥ 1. We also give some formulas for the sum of x?andy?coordinates of the points (x, y) on Ep,t. In the third section weconsider the rank of Et: y2= x3- t2x and its 2-isogenous curve Et over Q. We proved that the rank of Etand Etis 2 over Q. In the last section we obtain some formulas for the sums Σt∈F?panp,t for an integer n ≥ 1, where ap,t denote the trace of Frobenius.

Keywords: Elliptic curves over finite fields, rational points onelliptic curves, rank, trace of Frobenius.

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1895


[1] A.O.L. Atkin and F. Moralin. Eliptic Curves and Primality Proving.Math. Comp. 61 (1993), 29?68.
[2] A. Dujella. An Example of Elliptic Curve over Q with Rank Equal to15. Proc. Japan Acad. Ser. A Math. Sci. 78(2002), 109?111.
[3] N.D. Elkies. Some More Rank Records: E(Q) = (Z/2Z) ? Z18,(Z/4Z)?Z12, (Z/8Z)?Z6, (Z/2Z)? (Z/6Z)?Z6. Number TheoryListserver, Jun 2006.
[4] S. Fermigier. Exemples de Courbes Elliptiques de Grand Rang sur Q(t)et sur Q Possedant des points d?ordre 2. C.R. Acad. Sci. Paris Ser. I322(1996), 949?952.
[5] S. Goldwasser and J. Kilian. Almost all Primes can be Quickly Certified.In Proc. 18th STOC, Berkeley, May 28-30, 1986, ACM, New York(1986), 316-329.
[6] N. Koblitz. A Course in Number Theory and Cryptography. Springer-Verlag, 1994.
[7] T.J. Kretschmer. Construction of Elliptic Curves with Large Rank. Math.Comp. 46 (1986), 627?635.
[8] F. Lemmermeyer and R.A. Mollin. On the Tate-Shafarevich Groups ofy2 = x(x2 ? k2). Acta Math. Universitatis Comenianae LXXII(1)(2003), 73?80.
[9] H.W.Jr. Lenstra. Factoring Integers with Elliptic Curves. Annals ofMaths. 126(3) (1987), 649?673.
[10] R. Martin and W. McMillen. An Elliptic Curve Over Q with Rank atleast 24. Number Theory Listserver, May 2000.
[11] V.S. Miller. Use of Elliptic Curves in Cryptography, in Advances inCryptology?CRYPTO?85. Lect. Notes in Comp. Sci. 218, Springer-Verlag, Berlin (1986), 417?426.
[12] R.A. Mollin. An Introduction to Cryptography. Chapman&Hall/CRC,2001.
[13] L.J. Mordell. On the Rational Solutions of the Indeterminate Eqnarraysof the Third and Fourth Degrees. Proc. Cambridge Philos. Soc. 21(1922),179?192.
[14] U. Schneiders and H.G. Zimmer. The Rank of Elliptic Curves uponQuadratic Extensions, in: Computational Number Theory. (A. Petho,H.C. Williams, H.G. Zimmer, eds.), de Gruyter, Berlin, 1991.
[15] R. Schoof. Counting Points on Elliptic Curves Over Finite Fields.Journal de Theorie des Nombres de Bordeaux 7(1995), 219?254.
[16] A. Tekcan. The Elliptic Curves y2 = x(x ? 1)(x ? ?). Accepted byArs Combinatoria.
[17] J.H. Silverman. The Arithmetic of Elliptic Curves. Springer-Verlag, 1986.
[18] L.C. Washington. Elliptic Curves, Number Theory and Cryptography.Chapman&Hall /CRC, Boca London, New York, Washington DC, 2003.
[19] A. Wiles. Modular Elliptic Curves and Fermat?s Last Theorem. Annalsof Maths. 141(3) (1995), 443?551.