Primality of the number of points on an elliptic curve over a finite field

Abstract

Given a fixed elliptic curve E defined over Q having no rational torsion points, we discuss the probability that the number of points on E mod p is prime as the prime p varies. We give conjectural asymptotic formulas for the number of p ≤ n for which this number is prime, both in the case of a complex multiplication and a non-CM curve E. Numerical evidence is given supporting these formulas. © 1988 by Pacific Journal of Mathematics.