A REFINEMENT OF KOBLITZ'S CONJECTURE

Abstract

Let E be an elliptic curve over the rationals. In 1988, Koblitz conjectured an asymptotic for the number of primes p for which the cardinality of the group of 𝔽p-points of E is prime. However, the constant occurring in his asymptotic does not take into account that the distributions of the |E(𝔽p)| need not be independent modulo distinct primes. We shall describe a corrected constant. We also take the opportunity to extend the scope of the original conjecture to ask how often |E(𝔽p)|/t is an integer and prime for a fixed positive integer t, and to consider elliptic curves over arbitrary number fields. Several worked out examples are provided to supply numerical evidence for the new conjecture.