Hecke reciprocity and class groups

Abstract: When the $C_3$-extensions $F = K(\sqrt[3]{n})$ of $K = \mathbb{Q}(\sqrt{-3})$ are ordered by the norm of $n \in O_K$, we show that the average size of $\mathrm{Cl}F[2]$ is $3/2$, confirming a prediction of Cohen--Martinet. Over $\mathbb{Q}$, we prove that as $F$ varies over the pure cubic fields $\mathbb{Q}(\sqrt[3]{n})$ that are tamely (resp.\ wildly) ramified at $3$, the average size of $\mathrm{Cl}_F[2]$ is $3/2$ (resp.\ $2$). This tame/wild dichotomy is not accounted for by the Cohen--Lenstra type heuristics in the literature. Underlying this phenomenon is a reciprocity law for the relative class group $\mathrm{Cl}{F/K}[2]$ of odd degree extensions of number fields $F/K$. This leads to a random model for $\mathrm{Cl}{F/K}[2]$ that recovers Malle's heuristics for $\mathrm{Cl}{F/K}[2]$ in families of $K$-extensions with fixed Galois group. More generally, we formulate a heuristic for families with a fixed Galois $K$-group that explains the aberrant behavior in the family $\mathbb{Q}(\sqrt[3]{n})$ and predicts similar behavior in other special families. The other main ingredient in our proof is the work of Alp\"oge--Bhargava--Shnidman, on the number of integral $G(\mathbb{Q})$-orbits in a $G$-invariant quadric with bounded invariants.