Hecke reciprocity and class groups

We compute the average size of ${\mathrm{C}\mathrm{l}}_F[2]$ in the family of cubic fields $F=\mathbb{Q}(\sqrt[3]{n})$. Specifically, as $F$ varies over the subfamily of wildly (resp. tamely) ramified fields $\mathbb{Q}(\sqrt[3]{n})$, the average size of ${\mathrm{C}\mathrm{l}}_F[2]$ is $3\mathrm{/}2$ (resp. $2$). This tame/wild dichotomy is not accounted for by the class group heuristics in the literature. Analogously, when the extensions $F=K(\sqrt[3]{n})$ of $K=\mathbb{Q}(\sqrt{-3})$ are ordered by the norm of $n\in {\mathcal{O}}_K$, we show that the average size of ${\mathrm{C}\mathrm{l}}_F[2]$ is $3\mathrm{/}2$, as is predicted by the Cohen–Martinet heuristics for $C_3$-extensions of $K$. Underlying our proofs is a reciprocity law for the relative class groups ${\mathrm{C}\mathrm{l}}_{F\mathrm{/}K}[2]$ of odd degree extensions of number fields $F\mathrm{/}K$. This leads us to propose class group heuristics for families of $K$-extensions 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 is the work of Alpöge–Bhargava–Shnidman on the number of integral $G(\mathbb{Q})$-orbits in a $G$-invariant quadric with bounded invariants.