Curves of genus two with maps of every degree to a fixed elliptic curve
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913177336283136 |
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| author | Howe, Everett W. |
| author_facet | Howe, Everett W. |
| contents | We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19050 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Curves of genus two with maps of every degree to a fixed elliptic curve Howe, Everett W. Number Theory Algebraic Geometry 14H45 (Primary) 11G05, 11G10, 11G15, 11G30 (Secondary) We show that up to isomorphism there are exactly twenty pairs $(C,E)$, where $C$ is a genus-$2$ curve over ${\mathbf C}$, where $E$ is an elliptic curve over ${\mathbf C}$, and where for every integer $n>1$ there is a map of degree $n$ from $C$ to $E$. We also show that for every genus-$2$ curve $C$, there is an integer $n$ with $1 < n \le 59$ such that there is no minimal degree-$n$ map from $C$ to an elliptic curve. |
| title | Curves of genus two with maps of every degree to a fixed elliptic curve |
| topic | Number Theory Algebraic Geometry 14H45 (Primary) 11G05, 11G10, 11G15, 11G30 (Secondary) |
| url | https://arxiv.org/abs/2601.19050 |