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

TL;DR

This paper studies the distribution of the number of prime factors of elliptic curve groups over finite fields, providing asymptotic formulas for extreme values and focusing on CM elliptic curves.

Contribution

It establishes precise asymptotic counts for the number of primes where the prime factor count of elliptic curve groups exceeds or falls below certain thresholds, especially for CM curves.

Findings

Asymptotic formulas for primes with many prime factors in elliptic curve groups

Results apply to both high and low prime factor counts with explicit asymptotics

Method can be adapted to other CM elliptic curves

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.