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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| Accès en ligne: | https://arxiv.org/abs/2509.24937 |
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| _version_ | 1866909814801563648 |
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| author | Koymans, Peter Morgan, Adam |
| author_facet | Koymans, Peter Morgan, Adam |
| contents | Let $K$ be a number field, let $g \geq 1$ be an integer and let $f(x) = (x - a_1) \cdots (x - a_{2g + 1}) \in O_K[x]$ be a polynomial that splits into $2g + 1$ distinct linear factors. Write $C$ for the hyperelliptic curve given by $C: y^2 = f(x)$ and write $J = \mathrm{Jac}(C)$ for its Jacobian. Under mild technical assumptions on $f$ that are satisfied almost always, we prove that there exists some $d \in K^\times$ such that the quadratic twist $J^d$ has rank exactly equal to $1$. As a consequence, we deduce that for any positive integer $g$, there exists an absolutely simple abelian variety over $K$ with dimension equal to $g$ and rank equal to $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24937 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Constructing Jacobians of rank 1 Koymans, Peter Morgan, Adam Number Theory Let $K$ be a number field, let $g \geq 1$ be an integer and let $f(x) = (x - a_1) \cdots (x - a_{2g + 1}) \in O_K[x]$ be a polynomial that splits into $2g + 1$ distinct linear factors. Write $C$ for the hyperelliptic curve given by $C: y^2 = f(x)$ and write $J = \mathrm{Jac}(C)$ for its Jacobian. Under mild technical assumptions on $f$ that are satisfied almost always, we prove that there exists some $d \in K^\times$ such that the quadratic twist $J^d$ has rank exactly equal to $1$. As a consequence, we deduce that for any positive integer $g$, there exists an absolutely simple abelian variety over $K$ with dimension equal to $g$ and rank equal to $1$. |
| title | Constructing Jacobians of rank 1 |
| topic | Number Theory |
| url | https://arxiv.org/abs/2509.24937 |