Cohen–Lenstra heuristics for $3$-torsion in quadratic fields

For an odd prime $p$ and a positive integer $\alpha$, let $\operatorname{Conj}^{+}(p,\alpha)$ and $\operatorname{Conj}^{-}(p,\alpha)$ denote the stated Cohen–Lenstra average-value conjectures for real and imaginary quadratic fields, respectively. In particular, $\operatorname{Conj}^{+}(3,\alpha)$ and $\operatorname{Conj}^{-}(3,\alpha)$ concern the corresponding averages of

$$\prod_{0\leq i<\alpha}(h_3(K)-3^i),$$

where $h_3(K)$ is the size of the $3$-torsion subgroup of the class group of $K$.

Cohen–Lenstra conjecture for $p=3$. $\operatorname{Conj}^{+}(3,\alpha)$ and $\operatorname{Conj}^{-}(3,\alpha)$ are true for every positive integer $\alpha$.

These conjectures are known for $p=3$ and $\alpha=1$, but remain open in the other cases described by the source. The paper uses them as hypotheses to improve upper bounds for counting algebraic tori over $\mathbb{Q}$.