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

In this paper, we investigate extreme values of $\omega (E({\mathbb{F}}_p))$, where $E\mathrm{/}\mathbb{Q}$ is an elliptic curve with complex multiplication and $\omega$ is the number-of-distinct-prime-divisors function. For fixed $\gamma > 1$, we prove that $$\mathrm{\#}\{p\le x:\omega (E({\mathbb{F}}_p))>\gamma \log \log x\}=\frac{x}{(\log x{)}^{2+\gamma \log \gamma -\gamma +o(1)}}\mathrm{.}$$ The same result holds for the quantity $#{p \leq x : \omega(E($\mathbb{F}$_p)) < \gamma\log\log x}$when$0 < \gamma < 1$. 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.