A parametrization of $3$-class groups of quadratic rings over Dedekind domains

Eliot Hodges; Ashvin A. Swaminathan

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arXiv · 2509.01722

A parametrization of $3$-class groups of quadratic rings over Dedekind domains

Eliot Hodges·Ashvin A. Swaminathan

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Abstract

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 $R$.

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Eliot Hodges, Ashvin A. Swaminathan. 2025-09-01. A parametrization of $3$-class groups of quadratic rings over Dedekind domains. https://arxiv.org/abs/2509.01722

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