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

Let $R$ be a Dedekind domain with field of fractions $K$ and $\mathrm{char}(R)\mathrm{\ne }3$. 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 ${\mathrm{GL}}_2(K)$ whose actions on certain lattices of binary cubic forms over $K$ parametrize $3$-torsion ideal classes in class groups of quadratic rings over $R$.