The frequency of elliptic curve groups over prime finite fields

TL;DR

This paper investigates how often specific elliptic curve groups over prime finite fields occur, providing new bounds and asymptotic results without relying on unproven conjectures.

Contribution

It establishes unconditional bounds for the frequency of elliptic curve groups and confirms the conjectured asymptotic behavior for most groups under certain conditions.

Findings

Bounded the frequency $M(G_{m,k})$ by a constant multiple of the expected value when $m extless=k^A

Proved the asymptotic formula for $M(G_{m,k})$ holds for almost all groups when $m extless=k^{1/4- ext{epsilon}}$

Applied methods to study the frequency of integers as group orders of elliptic curves

Abstract

Letting $p$ vary over all primes and $E$ vary over all elliptic curves over the finite field $\mathbb{F}_p$, we study the frequency to which a given group $G$ arises as a group of points $E(\mathbb{F}_p)$. It is well-known that the only permissible groups are of the form $G_{m,k}:=\mathbb{Z}/m\mathbb{Z}\times \mathbb{Z}/mk\mathbb{Z}$. Given such a candidate group, we let $M(G_{m,k})$ be the frequency to which the group $G_{m,k}$ arises in this way. Previously, the second and fourth named authors determined an asymptotic formula for $M(G_{m,k})$ assuming a conjecture about primes in short arithmetic progressions. In this paper, we prove several unconditional bounds for $M(G_{m,k})$, pointwise and on average. In particular, we show that $M(G_{m,k})$ is bounded above by a constant multiple of the expected quantity when $m\le k^A$ and that the conjectured asymptotic for $M(G_{m,k})$ holds for almost all groups $G_{m,k}$ when $m\le k^{1/4-\epsilon}$. We also apply our methods to study the frequency to which a given integer $N$ arises as the group order $\#E(\mathbb{F}_p)$.