Density of discriminants of cubic extensions. (English) Zbl 0632.12007

The authors generalize the theorems of Davenport and Heilbronn on the density of discriminants of cubic fields to the case of cubic extensions of an arbitrary global field of characteristic not 2 or 3. The density of discriminants of all cubic extensions of such a given base field k is calculated. The partial densities assuming that the extensions satisfy finitely many localization conditions at places of k are also computed. Finally, as F ranges over all the quadratic extensions of k, the mean value of the number of ideal classes in F with both cube and relative norm equal to the identity class in F is determined. The methods differ from those of Davenport and Heilbronn in that the starting point is taken to be Shintani’s theory of the zeta-function associated with the space of binary cubic forms, in the adelic formulation developed in earlier papers of the authors.


11R16 Cubic and quartic extensions
11R45 Density theorems
11E12 Quadratic forms over global rings and fields
11R80 Totally real fields
11R56 Adèle rings and groups
11R42 Zeta functions and \(L\)-functions of number fields
11S40 Zeta functions and \(L\)-functions
11E76 Forms of degree higher than two
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