Ideal class monoids of cubic orders

Abstract: 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 $R$.

• The paper obtains an Euler product for the genus set of cubic orders, equipped with explicit formulas for counting ideal classes.
• It explains that cubic overorders form more intricate hierarchical structures than Bass orders, requiring parametrization by Serre invariants and conductor exponents.
• The techniques are applied to demonstrate Bhargava's parametrization of balanced pairs for 2x3x3 integral cubes, providing a fixed-ring orbit count.

This paper by Cho, Hong, and Lee computes the ideal class monoid $\overline{\mathrm{Cl}(R)}$ of cubic orders and, in particular, obtains an exact Euler product for the genus set $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$ when $R$ is Gorenstein (2607.07063). The work extends the authors' earlier treatment of Bass orders (Cho et al., 2024) to the genuinely non-Bass setting: while overorders of a local Bass order form a chain, cubic overorders form branched posets governed by two parameters (the Serre invariant $S(R_v)$ and conductor exponents), so the enumeration is substantially more involved.

Global framework and reduction

The paper works with orders in étale $F$-algebras of degree 3 over a number field or local field. The key structural facts are:

• A product formula identifies $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$ with $\prod_v \overline{\mathrm{Cl}(R_v)}$, so the global count reduces to finitely many primes where $R_v$ is not maximal.
• A stratification over overorders $O'$ gives $$\#\overline{\mathrm{Cl}(R)}=\sum_{O'}\#\mathrm{cl}(O')$$, and a central lemma shows that for any cubic overorder $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$0, with equality exactly when $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$1 is non-Gorenstein (Cohen–Macaulay type 2). Hence:

$\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$2

Thus everything reduces to explicitly enumerating overorders and identifying which are Gorenstein.

A useful characterization proved early on: an order $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$3 of degree 3 is Gorenstein iff $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$4. Moreover every simple extension of $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$5 is Gorenstein, and conversely a Gorenstein order is a simple extension except in one exceptional case ($\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$6, $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$7), corresponding to the primitive binary cubic form $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$8 vanishing on all of $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$9. This equivalence is specific to degrees 2 and 3.

Irreducible case

For $R$0 a cubic field extension, treated separately in unramified and totally ramified cases, the paper introduces normal forms $R$1 and $R$2, parametrized by Serre invariant $R$3, conductor exponent $R$4, and a parameter $R$5 defined modulo explicit powers of $R$6. Inclusion relations among these normal forms are given by congruence conditions on the $R$7-parameters, and Gorensteinness is detected by a single linear relation: in the unramified case $R$8, in the totally ramified case $R$9.

From these classifications, Theorems give closed formulas for $S(R_v)$0 depending only on $S(R_v)$1 — piecewise rational functions in $S(R_v)$2 with denominators $S(R_v)$3, covering four parity cases each. Notably these hold for arbitrary (not necessarily monogenic) local cubic orders. Under the Gorenstein hypothesis, where $S(R_v)$4 and $S(R_v)$5 become linearly related ($S(R_v)$6, $S(R_v)$7 unramified; analogous relations totally ramified), the sums collapse to compact polynomial factors. For instance, in the totally ramified Gorenstein case $S(R_v)$8,

$S(R_v)$9

when $F$0 is odd, with a parallel even-$F$1 expression. Worked Hasse diagrams illustrate the branching: e.g. $F$2 with $F$3 exhibits multiple incomparable branches, contrasting sharply with the chain structure of Bass orders of the same Serre invariant.

Split case

Three splitting types $F$4, $F$5, and $F$6 are handled via uniform normal forms $F$7 (with type-specific ring-closure conditions). For each type the paper determines:

• the exact range of $F$8 giving orders;
• inclusion criteria between normal forms;
• Serre invariants and conductors as explicit functions of $F$9;
• Gorenstein criteria, again reducing to linear relations (e.g., $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$0 or variants involving $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$1).

The resulting counting formulas are multi-summations in the conductor components $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$2 and $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$3, expressed entirely by invariants; under the Gorenstein constraint these again evaluate to closed polynomial expressions. In the $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$4 Gorenstein case, the conductor must have two equal components with the third smallest and even, and $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$5 takes the compact form $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$6 with $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$7.

Global Gorenstein formula and Bhargava application

Combining all local factors yields the main theorem: for a Gorenstein cubic order $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$8 with conductor $\mathrm{Cl}(R)\backslash\overline{\mathrm{Cl}(R)}$9 and splitting types $\prod_v \overline{\mathrm{Cl}(R_v)}$0 at primes above the conductor,

$\prod_v \overline{\mathrm{Cl}(R_v)}$1

where each $\prod_v \overline{\mathrm{Cl}(R_v)}$2 is an explicit polynomial in $\prod_v \overline{\mathrm{Cl}(R_v)}$3 determined solely by the local conductor tuple. This is an exact genus count, not merely asymptotic.

As an application to Bhargava's parametrization of $\prod_v \overline{\mathrm{Cl}(R_v)}$4 cubes, the paper proves that for Gorenstein $\prod_v \overline{\mathrm{Cl}(R_v)}$5, balanced pairs $\prod_v \overline{\mathrm{Cl}(R_v)}$6 are always equivalent to $\prod_v \overline{\mathrm{Cl}(R_v)}$7 — using local duality to show $\prod_v \overline{\mathrm{Cl}(R_v)}$8 forces equality from the inclusion $\prod_v \overline{\mathrm{Cl}(R_v)}$9 — so integral orbits with cubic ring $R_v$0 are identified with $R_v$1. Consequently the number of $R_v$2-equivalence classes (genera) of such orbits is given exactly by the Euler product above. This complements the usual asymptotic orbit-counting philosophy (Bhargava, Sato–Shintani) by fixing the ring rather than averaging over it.

Two further uses of the local enumeration are noted. First, it supplies the input that makes the functional equation for the $R_v$3-function of a Gorenstein cubic order unconditional in work on Beyond Endoscopy for $R_v$4 (Lee, 3 Jul 2026), removing a conjectural factorization assumption in the Deng–Espinosa program. Second, the overorder classification provides finite local data for Hitchin-theoretic extensions of the Bass-case geometrization of the authors' companion work.

Limitations and open questions

Several caveats are stated plainly. The closed Euler product requires the Gorenstein hypothesis; for non-Gorenstein local orders the counts remain finite and computable but do not collapse to conductor-controlled polynomials, and the paper omits an explicit global closed formula in that generality, providing instead Python algorithms evaluating the split-case multi-summations. The identification of balanced pairs with the ideal class monoid fails without Gorensteinity: the dual $R_v$5 need not have inverse norm, and the fixed-ring orbit problem then depends on more than $R_v$6. The equivalence "Gorenstein = simple extension" also has the small-residue-field exception noted above, and does not extend beyond degree 3. Open directions left by the paper include the geometric/stack-theoretic interpretation of the non-Gorenstein contribution via the Artin stack of Wood balanced pairs (a sequel in preparation) and the cubic analogue of the Hitchin fibration strata.

Conclusion

The paper completes a program begun for Bass orders by giving, for arbitrary local cubic orders, a full parametrization of overorders with their inclusion relations and Gorenstein loci, and deriving from this an exact Euler product for the genus set of Gorenstein cubic orders together with an exact fixed-ring orbit count for Bhargava's $R_v$7 cubes. The results are purely algebraic, depend only on conductor data, and serve both arithmetic applications (Beyond Endoscopy) and geometric developments built on the same local classification.