Poisson distribution of a prime counting function corresponding to elliptic curves

TL;DR

This paper demonstrates that the distribution of the number of primes for which the group of points on an elliptic curve over finite fields has a fixed order follows a Poisson distribution, on average over many curves.

Contribution

It establishes the Poisson distribution law for the prime counting function associated with elliptic curves over finite fields, averaged over a broad class of curves.

Findings

$M_E(N)$ follows a Poisson distribution on average

Distribution results hold over a large class of elliptic curves

Provides probabilistic insight into elliptic curve point counts

Abstract

Let $E$ be an elliptic curve defined over rational field $\mathbb{Q}$ and $N$ be a positive integer. Now, $M_E(N)$ denotes the number of primes $p$, such that the group $E_p(\mathbb{F}_p)$ is of order $N$. We show that $M_E(N)$ follows Poisson distribution when an average is taken over a large class of curves.