Density of discriminants of cubic extensions.

*(English)*Zbl 0632.12007The 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.

##### MSC:

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 |