Orders of reductions of elliptic curves with many and few prime factors

In this paper, we investigate extreme values of omega(# E(F-p)), where E/Q is an elliptic curve with complex multiplication and w is the number-of-distinct-prime-divisors function. For fixed omega > 1, we prove an asymptotic formula for the quantity #{p <= x : omega(# E(F-p)) > xi log log x}. The same result holds for the quantity #{p = x :omega(# E(F-p)) < xi log log x} when 0 < xi < 1. This asymptotic formula matches what one might expect, based on a result of Delange concerning extreme values of omega(n). The argument is worked out in detail for the curve E : y(2) = x(3) - x, and we discuss how the method can be adapted for other CM elliptic curves.