Cohen–Martinet conjecture for relative class groups

Let $\Sigma=(H,K_0,\sigma)$ be a situation satisfying the stated character-theoretic assumptions, let ${\mathcal{O}}={\mathcal{O}}(\Sigma)$, and let ${\mathfrak{p}}$ be a prime ideal of ${\mathcal{O}}$. For a finite ${\mathfrak{p}}$-torsion ${\mathcal{O}}$-module $G$, let ${\mathcal{N}}(\Sigma,G)$ be the limiting proportion of fields $K\in{\mathcal{K}}(\Sigma)$ whose relative class group has ${\mathfrak{p}}$-part isomorphic to $G$, ordered by relative discriminant. Cohen–Martinet conjecture. The limit ${\mathcal{N}}(\Sigma,G)$ exists and is

$$\frac{(q)_\infty}{(q)_u}\cdot\frac{1}{|G|^u|{\operatorname{Aut}}_{\mathcal{O}}(G)|},$$

where $u=u(\Sigma)$, $q=|{\mathcal{O}}/{\mathfrak{p}}|$, and ${\operatorname{Aut}}_{\mathcal{O}}(G)$ is the group of ${\mathcal{O}}$-automorphisms of $G$. This generalizes the Cohen–Lenstra heuristic from imaginary quadratic fields to relative class groups in broader families of number fields; the conjecture predicts both existence and an explicit limiting distribution.