Cohen–Lenstra conjecture for Sylow subgroups of quadratic class groups

Let $S_X^+$ (resp. $S_X^-$) denote the set of positive (resp. negative) fundamental discriminants $D$ with $|D|<X$. Let $p$ be an odd prime and let $B$ be a finite abelian $p$-group.

Cohen–Lenstra conjecture. The proportion of discriminants $D\in S_X^{\pm}$ for which the Sylow $p$-subgroup of the class group of $\mathbb{Q}(\sqrt D)$ is isomorphic to $B$ satisfies

$$\lim_{X\to\infty}\frac{\#\{D\in S_X^{\pm}:\operatorname{Cl}(\mathbb{Q}(\sqrt D))_p\simeq B\}}{|S_X^{\pm}|}=\frac{\prod_{k=1}^{\infty}(1-p^{-k-u})}{|B|^u|\operatorname{Aut}(B)|},$$

where $u=0$ for $S_X^-$ and $u=1$ for $S_X^+$.

This heuristic predicts the distribution of Sylow $p$-subgroups of class groups of quadratic number fields and is the Cohen–Lenstra distribution approached by cokernels of suitable large random integral matrices. The source does not provide evidence resolving the conjecture itself.