Ideal class monoids of cubic orders

Let $R$ be an order in a number field, let $\overline{\mathrm{Cl}}(R)$ be its ideal class monoid, and let $\mathrm{Cl}(R)$ act on it by multiplication.

The local-global product formula identifies the orbit set $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}}(R)$ with a product of local orbit sets; in this sense, it is the genus set of fractional $R$-ideals.

For a Gorenstein order $R$ in a cubic extension of number fields, we give a closed Euler product formula for the cardinality of this genus set.

The local factors come from an explicit classification of local cubic overorders: for arbitrary local cubic orders, we parametrize all overorders, determine their inclusion relations, and identify the Gorenstein ones.

As an application to Bhargava's parametrization of $2\times3\times3$ cubes, our formula gives the exact number of $\mathrm{Cl}(R)$-equivalence classes of integral $\mathrm{GL}_2(\mathbb Z)\times\mathrm{SL}_3(\mathbb Z)\times\mathrm{SL}_3(\mathbb Z)$-orbits whose associated cubic ring is the prescribed Gorenstein order $R$.