On the mean number of 2 -torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields

Abstract

Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of $2$-torsion elements in the class groups and narrow class groups of these cubic fields when they are ordered by their absolute discriminants. For an order $\mathcal{O}$ in a cubic field, we study three groups: ${\mathrm{Cl}}_2\left(\mathcal{O}\right)$, the group of ideal classes of $\mathcal{O}$ of order $2$; ${\mathrm{Cl}}_2^{+}\left(\mathcal{O}\right)$, the group of narrow ideal classes of $\mathcal{O}$ of order $2$; and ${\mathcal{I}}_2\left(\mathcal{O}\right)$, the group of ideals of $\mathcal{O}$ of order $2$. We prove that the mean value of the difference $\left|{\mathrm{Cl}}_2\right(\mathcal{O}\left)\right|-\frac{1}{4}\left|{\mathcal{I}}_2\right(\mathcal{O}\left)\right|$ is always equal to $1$, regardless of whether one averages over the maximal orders in real cubic fields, over all orders in real cubic fields, or indeed over any family of real cubic orders defined by local conditions. For the narrow class group, we prove that the average value of the difference $\left|{\mathrm{Cl}}_2^{+}\right(\mathcal{O}\left)\right|-\left|{\mathcal{I}}_2\right(\mathcal{O}\left)\right|$ is equal to $1$ for any such family. Also, for any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference $\left|{\mathrm{Cl}}_2\right(\mathcal{O}\left)\right|-\frac{1}{2}\left|{\mathcal{I}}_2\right(\mathcal{O}\left)\right|$ is always equal to $1$, independent of the family. The determination of these mean numbers allows us to prove a number of further results as by-products. Most notably, we prove—in stark contrast to the case of quadratic fields—that (1) a positive proportion of cubic fields have odd class number, (2) a positive proportion of real cubic fields have isomorphic $2$-torsion in the class group and the narrow class group, and (3) a positive proportion of real cubic fields contain units of mixed real signature. We also show that a positive proportion of real cubic fields have narrow class group strictly larger than the class group, and thus a positive proportion of real cubic fields do not possess units of every possible real signature.