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arXiv · 2411.13526

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

Adrian Barquero-Sanchez·Jimmy Calvo-Monge
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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.

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Adrian Barquero-Sanchez, Jimmy Calvo-Monge. 2024-11-20. The density and distribution of CM elliptic curves over $\mathbb{Q}$. https://arxiv.org/abs/2411.13526

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