Hecke reciprocity and class groups

Abstract

We compute the average size of $\mathrm{Cl}_F[2]$ 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 $\mathrm{Cl}_F[2]$ is $3/2$ (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 $\mathrm{Cl}_F[2]$ is $3/2$, 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 $\mathrm{Cl}_{F/K}[2]$ of odd degree extensions of number fields $F/K$. 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 Alp\"oge--Bhargava--Shnidman on the number of integral $G(\mathbb{Q})$-orbits in a $G$-invariant quadric with bounded invariants.