  Let E be an elliptic curve over the number field Q. In 1988, Koblitz
conjectured an asymptotic for the number of primes p for which the cardinality
of the group of F_p-points of E is prime. However, the constant occurring in
his asymptotic does not take into account that the distributions of the
|E(F_p)| 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(F_p)|/t is 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.
