prime divisors of the number of rational points on elliptic curves with complex multiplication

Abstract

let $e/\mathbb{q}$ be an elliptic curve. for a prime $p$ of good reduction, let $e(\mathbb{f}_p)$ be the set of rational points defined over the finite field $\mathbb{f}_p$. denote by $\omega(\#e(\mathbb{f}_p))$ the number of distinct prime divisors of $\#e(\mathbb{f}_p)$. for an elliptic curve with complex multiplication, the normal order of $\omega(\#e(\mathbb{f}_p))$ is shown to be $\log \log p$. the normal order of the number of distinct prime factors of the exponent of $e(\mathbb{f}_p)$ is also studied.