## 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.

## Citation

Manjul Bhargava. Ila Varma. "On the mean number of $2$-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields." Duke Math. J. 164 (10) 1911 - 1933, 15 July 2015. https://doi.org/10.1215/00127094-3120636

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