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

Abstract

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