A parametrization of $3$-class groups of quadratic rings over Dedekind domains
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| Format: | Preprint |
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2025
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| _version_ | 1866908514656452608 |
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| author | Hodges, Eliot Swaminathan, Ashvin A. |
| author_facet | Hodges, Eliot Swaminathan, Ashvin A. |
| contents | Let $R$ be a Dedekind domain with field of fractions $K$ and $\operatorname{char}(R)\neq3$. In this paper, we generalize Bhargava's parametrization of $3$-torsion ideal classes by binary cubic forms to work over $R$. Specifically, we construct arithmetic subgroups of $\operatorname{GL}_2(K)$ whose actions on certain lattices of binary cubic forms over $K$ parametrize $3$-torsion ideal classes in class groups of quadratic rings over $R$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_01722 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A parametrization of $3$-class groups of quadratic rings over Dedekind domains Hodges, Eliot Swaminathan, Ashvin A. Number Theory 11E76, 11E12 (Primary) 11R11, 11R29, 13C20 (Secondary) Let $R$ be a Dedekind domain with field of fractions $K$ and $\operatorname{char}(R)\neq3$. In this paper, we generalize Bhargava's parametrization of $3$-torsion ideal classes by binary cubic forms to work over $R$. Specifically, we construct arithmetic subgroups of $\operatorname{GL}_2(K)$ whose actions on certain lattices of binary cubic forms over $K$ parametrize $3$-torsion ideal classes in class groups of quadratic rings over $R$. |
| title | A parametrization of $3$-class groups of quadratic rings over Dedekind domains |
| topic | Number Theory 11E76, 11E12 (Primary) 11R11, 11R29, 13C20 (Secondary) |
| url | https://arxiv.org/abs/2509.01722 |