Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866912231603568640 |
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| author | Narbonne, Fabien |
| author_facet | Narbonne, Fabien |
| contents | Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $\mathbb{C}$ with complex multiplication (CM) by $R$ and the category of integral hermitian $R$-lattices. Then we apply this equivalence to enumerate all the genus $2$ and $3$ curves with field of moduli $\mathbb{Q}$ and with Jacobian isomorphic to a product of elliptic curves with CM by $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_11982 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$ Narbonne, Fabien Number Theory Let $R$ be the maximal order in a quadratic imaginary field $K$. We give an equivalence of categories between the category of polarized abelian varieties isomorphic to a product of elliptic curves over $\mathbb{C}$ with complex multiplication (CM) by $R$ and the category of integral hermitian $R$-lattices. Then we apply this equivalence to enumerate all the genus $2$ and $3$ curves with field of moduli $\mathbb{Q}$ and with Jacobian isomorphic to a product of elliptic curves with CM by $R$. |
| title | Polarized products of elliptic curves with complex multiplication and field of moduli $\mathbb{Q}$ |
| topic | Number Theory |
| url | https://arxiv.org/abs/2203.11982 |