Primality of the number of points on an elliptic curve over a finite field
N Koblitz - Pacific journal of mathematics, 1988 - msp.org
Pacific journal of mathematics, 1988•msp.org
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.
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.
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.
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