Almost prime orders of CM elliptic curves modulo p

Given an elliptic curve over Q with complex multiplication by OK, the ring of integers of the quadratic imaginary field K, we analyze the integer d(E) =gcd{|E(F-p)| : p splits in O-K}, where |E(F-p)| is the size of the group of rational IF points, and prove that it can be bigger than the common factor that comes from the torsion of the curve. Then, we prove that # {p <= x, p splits in O-K : 1/d(E) |E(F-p)| = P-2} >> x/(log x)(2) hence extending the results in [16]. This is the best known result in the direction of the Koblitz conjecture about the primality of |E(F-p)|.