Classification of the Bachet Elliptic Curves y2 = x3 + a3 in Fp, where p ≡ 1 (mod 6) is Prime

In this work, we first give in what fields Fp, the cubic root of unity lies in F*p, in Qp and in K*p where Qp and K*p denote the sets of quadratic and non-zero cubic residues modulo p. Then we use these to obtain some results on the classification of the Bachet elliptic curves y2 ≡ x3 +a3 modulo p, for p ≡ 1 (mod 6) is prime.

Authors:

• Nazli Yildiz İkikardes
• Gokhan Soydan
• Musa Demirci
• Ismail Naci Cangul

Keywords:

• Elliptic curves over finite fields
• quadratic residue
• cubic residue.

References:

[1] Naml─▒, D., Cubic Residues , PhD Thesis, Balikesir University, (2001)
[2] Demirci, M. & Soydan, G. & Cang¨ul, I. N., Rational points on the
elliptic curves y2 = x3 + a3 (mod p) in Fp where p≡ 1(mod 6) is
prime, Rocky J. of Maths, (to be printed).
[3] Soydan, G. & Demirci, M. & Ikikardes┬©, N. Y. & Cang┬¿ul, I. N., Rational
points on the elliptic curves y2 = x3+a3 (mod p) in Fp where p≡ 5
(mod 6) is prime, (submitted).
[4] Silverman, J. H., The Arithmetic of Elliptic Curves, Springer-Verlag,
(1986), ISBN 0-387-96203-4.
[5] Silverman, J. H.,Tate, J., Rational Points on Elliptic Curves, Springer-
Verlag, (1992), ISBN 0-387-97825-9.
[6] Parshin, A. N., The Bogomolov-Miyaoka-Yau inequality for the arithmetical
surfaces and its applications, Seminaire de Theorie des Nombres,
Paris, 1986-87, 299-312, Progr. Math., 75, Birkhauser Boston, MA,
1998.
[7] Kamienny, S., Some remarks on torsion in elliptic curves, Comm. Alg.
23 (1995), no. 6, 2167-2169.
[8] Ono, K., Euler-s concordant forms, Acta Arith. 78 (1996), no. 2, 101-
123.
[9] Merel, L., Arithmetic of elliptic curves and Diophantine equations, Les
XXemes Journees Arithmetiques (Limoges, 1997), J. Theor. Nombres
Bordeaux 11 (1999), no. 1, 173-200.
[10] Serre, J.-P., Propri'et'es galoisiennes des points d-ordre fini des courbes
elliptiques, Invent. Math. 15 (1972), 259-331.