Finding the complement of an elliptic curve inside a Jacobian
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910282254647296 |
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| author | Gallese, Andrea Lombardo, Davide Naccarato, Francesco Zannier, Umberto |
| author_facet | Gallese, Andrea Lombardo, Davide Naccarato, Francesco Zannier, Umberto |
| contents | This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $π:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $π':X\to E'$ independent of $π$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02429 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finding the complement of an elliptic curve inside a Jacobian Gallese, Andrea Lombardo, Davide Naccarato, Francesco Zannier, Umberto Number Theory Algebraic Geometry 14H40, 14Q20 (Primary) 11G05, 14K02 (Secondary) This note gives a simple algorithm for the following effectivity problem: given a genus $2$ curve $X$ together with a nonconstant map $π:X\to E$ to an elliptic curve, determine an elliptic curve $E'$ and a map $π':X\to E'$ independent of $π$. Equivalently, we compute the complementary elliptic factor in the decomposition of $\operatorname{Jac}(X)$ up to isogeny. While the problem has been studied extensively, and more general ones have been solved by deep and powerful techniques, we are not aware of a reference for the simple explicit procedure described here. |
| title | Finding the complement of an elliptic curve inside a Jacobian |
| topic | Number Theory Algebraic Geometry 14H40, 14Q20 (Primary) 11G05, 14K02 (Secondary) |
| url | https://arxiv.org/abs/2606.02429 |