The density and distribution of CM elliptic curves over $\mathbb{Q}$

TL;DR

This paper investigates the distribution of CM elliptic curves over rationals, establishing that their density is zero when ordered by height, and showing that almost all have a specific CM order with $j$-invariant 0.

Contribution

It provides the first asymptotic formulas for counting CM elliptic curves by height and analyzes their distribution among CM orders, revealing a dominant order with $j=0$.

Findings

Density of CM elliptic curves over $Q$ is zero.

Almost all CM elliptic curves have $j$-invariant 0.

Asymptotic formulas for counting elliptic curves with fixed $j$-invariant.

Abstract

In this paper we study the density and distribution of CM elliptic curves over $\mathbb{Q}$. In particular, we prove that the natural density of CM elliptic curves over $\mathbb{Q}$, when ordered by naive height, is zero. Furthermore, we analyze the distribution of these curves among the thirteen possible CM orders of class number one. Our results show that asymptotically, $100\%$ of them have complex multiplication by the order $\mathbb{Z}\left[\frac{-1 + \sqrt{-3}}{2} \right]$, that is, have $j$-invariant 0. We conduct this analysis within two different families of representatives for the $\mathbb{Q}$-isomorphism classes of CM elliptic curves: one commonly used in the literature and another constructed using the theory of twists. As part of our proofs, we give asymptotic formulas for the number of elliptic curves with a given $j$-invariant and bounded naive height.