The density and distribution of CM elliptic curves over $\mathbb{Q}$
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866909397624553472 |
|---|---|
| author | Barquero-Sanchez, Adrian Calvo-Monge, Jimmy |
| author_facet | Barquero-Sanchez, Adrian Calvo-Monge, Jimmy |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_13526 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The density and distribution of CM elliptic curves over $\mathbb{Q}$ Barquero-Sanchez, Adrian Calvo-Monge, Jimmy Number Theory Algebraic Geometry 11G05 (Primary), 11G15, 11N45 (Secondary) 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. |
| title | The density and distribution of CM elliptic curves over $\mathbb{Q}$ |
| topic | Number Theory Algebraic Geometry 11G05 (Primary), 11G15, 11N45 (Secondary) |
| url | https://arxiv.org/abs/2411.13526 |