Orders of CM elliptic curves modulo p with at most two primes

TY - JOUR

T1 - Orders of CM elliptic curves modulo p with at most two primes

AU - Iwaniec, Henryk

AU - Urroz, Jorge Jiménez

PY - 2010

Y1 - 2010

N2 - In this paper of 1988 N. Koblitz conjectured that given an elliptic curve E over the rationals, the order of the group of Fp points of its reduction modulo p, |E(Fp)|, is a prime number for infinitely many primes p. Since then a wide number of research articles has been dedicated to understand and solve this conjecture. In this paper we give the best result known nowadays. We can prove quantitatively that for infinitely many primes p the reduction of the curve y2 = x3 - x modulo p has order which is eight times an almost prime number. The problem turns out to be the equivalent to the twin prime conjecture in the Gaussian domain. The result could be extended to any CM curve with certain considerations. We also point out the relation of the result with certain considerations. We also point out the relation of the result with the cyclicity of E(Fp), and the Lang Trotter conjecture.

AB - In this paper of 1988 N. Koblitz conjectured that given an elliptic curve E over the rationals, the order of the group of Fp points of its reduction modulo p, |E(Fp)|, is a prime number for infinitely many primes p. Since then a wide number of research articles has been dedicated to understand and solve this conjecture. In this paper we give the best result known nowadays. We can prove quantitatively that for infinitely many primes p the reduction of the curve y2 = x3 - x modulo p has order which is eight times an almost prime number. The problem turns out to be the equivalent to the twin prime conjecture in the Gaussian domain. The result could be extended to any CM curve with certain considerations. We also point out the relation of the result with certain considerations. We also point out the relation of the result with the cyclicity of E(Fp), and the Lang Trotter conjecture.

UR - http://www.scopus.com/inward/record.url?scp=83455213904&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=83455213904&partnerID=8YFLogxK

M3 - Article

SN - 0391-173X

VL - 9

SP - 815

EP - 832

JO - Annali della Scuola Normale - Classe di Scienze

JF - Annali della Scuola Normale - Classe di Scienze

IS - 4

ER -