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 \[ \#\{p \leq x : ω(E(\mathbb{F}_p)) > γ\log\log x\} = \frac{x}{(\log x)^{2 + γ\logγ- γ+ o(1)}}. \] The same result holds for the quantity $\#\{p\le x:\omega (E({\mathbb{F}}_p))<\gamma \log \log x\}$ 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.

arXiv comment15 pages