Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866916225139867648 |
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| author | Dallaporta, William |
| author_facet | Dallaporta, William |
| contents | Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any PID $R$. When ${R = \mathbb{K}[X]}$, we show that the Genus Theory map is the quadratic form version of the $2$-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of non-trivial specializations has density $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_13045 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes Dallaporta, William Number Theory Algebraic Geometry 11E16, 14H25, 14H40 (Primary), 11R45 (Secondary) Genus Theory is a classical feature of integral binary quadratic forms. Using the author's generalization of the well-known correspondence between quadratic form classes and ideal classes of quadratic algebras, we extend it to the case when quadratic forms are twisted and have coefficients in any PID $R$. When ${R = \mathbb{K}[X]}$, we show that the Genus Theory map is the quadratic form version of the $2$-descent map on a certain hyperelliptic curve. As an application, we make a contribution to a question of Agboola and Pappas regarding a specialization problem of divisor classes on hyperelliptic curves. Under suitable assumptions, we prove that the set of non-trivial specializations has density $1$. |
| title | Quadratic forms and Genus Theory : a link with 2-descent and an application to non-trivial specializations of ideal classes |
| topic | Number Theory Algebraic Geometry 11E16, 14H25, 14H40 (Primary), 11R45 (Secondary) |
| url | https://arxiv.org/abs/2210.13045 |