Odd and Even Elliptic Curves with Complex Multiplication
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908654989475840 |
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| author | Zarhin, Yuri G. |
| author_facet | Zarhin, Yuri G. |
| contents | We call an order $O$ in a quadratic field $K$ odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve $E$ over the field $C$ of complex numbers with CM odd (resp. even) if its endomorphism ring $End(E)$ is an odd (resp. even) order in the corresponding imaginary quadratic field.
Suppose that $j(E)$ is a real number and let us consider the set $J(R,E)$ of all $j(E')$ where $E'$ is any elliptic curve that enjoys the following properties.
1) $E'$ is isogenous to $E$;
2) $j(E')$ is a real number;
3) $E'$ has the same parity as $E$.
We prove that the closure of $J(R,E)$ in the set $R$ of real numbers is the closed semi-infinite interval $(-\infty,1728]$ (resp. the whole $R$) if $E$ is odd (resp. even).
This paper was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of $j$-invariants of certain elliptic curves of CM type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_07240 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Odd and Even Elliptic Curves with Complex Multiplication Zarhin, Yuri G. Number Theory Algebraic Geometry 11G07, 14H52 We call an order $O$ in a quadratic field $K$ odd (resp. even) if its discriminant is an odd (resp. even) integer. We call an elliptic curve $E$ over the field $C$ of complex numbers with CM odd (resp. even) if its endomorphism ring $End(E)$ is an odd (resp. even) order in the corresponding imaginary quadratic field. Suppose that $j(E)$ is a real number and let us consider the set $J(R,E)$ of all $j(E')$ where $E'$ is any elliptic curve that enjoys the following properties. 1) $E'$ is isogenous to $E$; 2) $j(E')$ is a real number; 3) $E'$ has the same parity as $E$. We prove that the closure of $J(R,E)$ in the set $R$ of real numbers is the closed semi-infinite interval $(-\infty,1728]$ (resp. the whole $R$) if $E$ is odd (resp. even). This paper was inspired by a question of Jean-Louis Colliot-Thélène and Alena Pirutka about the distribution of $j$-invariants of certain elliptic curves of CM type. |
| title | Odd and Even Elliptic Curves with Complex Multiplication |
| topic | Number Theory Algebraic Geometry 11G07, 14H52 |
| url | https://arxiv.org/abs/2406.07240 |