On the mean number of 2torsion 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 2torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $\rm Cl_2({\cal O})\frac14{\cal I}_2(\cal O)$ is always equal to $1$, 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 mean value of the difference $\rm Cl^+_2({\cal O}){\cal I}_2(\cal O)$ is equal to $1$ for any such family. For any family of complex cubic orders defined by local conditions, we prove similarly that the mean value of the difference $\rm Cl_2(\mathcal O)\frac12{\cal I}_2(\cal O)$ 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 byproducts. Most notably, we provein stark contrast to the case of quadratic fieldsthat: 1) a positive proportion of cubic fields have odd class number; 2) a positive proportion of real cubic fields have isomorphic 2torsion 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.
 Publication:

arXiv eprints
 Pub Date:
 February 2014
 arXiv:
 arXiv:1402.5738
 Bibcode:
 2014arXiv1402.5738B
 Keywords:

 Mathematics  Number Theory;
 11R29;
 11R45;
 11R16
 EPrint:
 17 pages