A curve and its abstract generalized Jacobian
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910210817261568 |
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| author | Castle, Benjamin Dan-Cohen, Ishai Hasson, Assaf |
| author_facet | Castle, Benjamin Dan-Cohen, Ishai Hasson, Assaf |
| contents | To a smooth proper curve $C$ over a field $k$ equipped with a $k$-point $c$ and an effective divisor $\mathfrak m$ coprime to $c$, one may associate the abstract group $J_{\mathfrak m}(\bar k)$ of $\overline k$-points of the generalized Jacobian, as well as a subset \[ \tag{*} \big(C\setminus \operatorname{Supp}(\mathfrak m)\big)(\bar k) \subset J_{\mathfrak m}(\bar k). \] We show that the data $(C,c,\mathfrak m)$ can be retrieved from (*) up to a twist by an automorphism of $\overline k$, proving a conjecture of Booher and Voloch. By a result of Booher and Voloch this shows that when $k$ is a finite field, the same data may also be retrieved from $L$-functions of characters of certain Galois extensions of the function field of $C$. The proof is a generalization of Zilber's well known work "A curve and its abstract Jacobian". |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09155 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A curve and its abstract generalized Jacobian Castle, Benjamin Dan-Cohen, Ishai Hasson, Assaf Algebraic Geometry Logic To a smooth proper curve $C$ over a field $k$ equipped with a $k$-point $c$ and an effective divisor $\mathfrak m$ coprime to $c$, one may associate the abstract group $J_{\mathfrak m}(\bar k)$ of $\overline k$-points of the generalized Jacobian, as well as a subset \[ \tag{*} \big(C\setminus \operatorname{Supp}(\mathfrak m)\big)(\bar k) \subset J_{\mathfrak m}(\bar k). \] We show that the data $(C,c,\mathfrak m)$ can be retrieved from (*) up to a twist by an automorphism of $\overline k$, proving a conjecture of Booher and Voloch. By a result of Booher and Voloch this shows that when $k$ is a finite field, the same data may also be retrieved from $L$-functions of characters of certain Galois extensions of the function field of $C$. The proof is a generalization of Zilber's well known work "A curve and its abstract Jacobian". |
| title | A curve and its abstract generalized Jacobian |
| topic | Algebraic Geometry Logic |
| url | https://arxiv.org/abs/2605.09155 |