The Number of Rational Points on Elliptic Curves y2 = x3 + a3 on Finite Fields
In this work, we consider the rational points on elliptic
curves over finite fields Fp. We give results concerning the number
of points Np,a on the elliptic curve y2 ≡ x3 +a3(mod p) according
to whether a and x are quadratic residues or non-residues. We use
two lemmas to prove the main results first of which gives the list of
primes for which -1 is a quadratic residue, and the second is a result
from [1]. We get the results in the case where p is a prime congruent
to 5 modulo 6, while when p is a prime congruent to 1 modulo 6,
there seems to be no regularity for Np,a.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:55974", author = "Musa Demirci and Nazlı Yıldız İkikardeş and Gökhan Soydan and İsmail Naci Cangül", title = "The Number of Rational Points on Elliptic Curves y2 = x3 + a3 on Finite Fields", abstract = "In this work, we consider the rational points on elliptic
curves over finite fields Fp. We give results concerning the number
of points Np,a on the elliptic curve y2 ≡ x3 +a3(mod p) according
to whether a and x are quadratic residues or non-residues. We use
two lemmas to prove the main results first of which gives the list of
primes for which -1 is a quadratic residue, and the second is a result
from [1]. We get the results in the case where p is a prime congruent
to 5 modulo 6, while when p is a prime congruent to 1 modulo 6,
there seems to be no regularity for Np,a.", keywords = "Elliptic curves over finite fields, rational points, quadratic residue.", volume = "1", number = "1", pages = "82-3", }