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On the mean number of 2-torsion elements in the class groups, narrow class groups, and ideal groups of cubic orders and fields
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 ordered by their absolute
discriminants.
For an order in a cubic field, we study the three groups: , the group of ideal classes of of order 2; , the group of narrow ideal classes of of order 2; and
, the group of ideals of of order 2. We prove that
the mean value of the difference is always equal to , 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 is equal to 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 is always equal to , 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.Comment: 17 page
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