The adelic zeta function associated to the space of binary cubic forms. II: Local theory.

*(English)*Zbl 0575.10016In this complement to an earlier paper of the second author [Math. Ann. 270, 503-534 (1985; Zbl 0533.10020)] the theory of zeta functions associated to the natural representation of \(Gl_ 2\) in the space of binary cubic forms over a local field is presented. The intimate connection between these local zeta functions and the arithmetic of quadratic and cubic extensions of a local field is completely revealed. Explicit evaluations of the zeta functions for p-fields are given. All the most interesting properties, including the functional equation, of the associated singular distributions are obtained.

In the last section, the authors synthesize these local results with the global theory published earlier to derive the basic properties of the generalization to arbitrary global fields of characteristic not 2 or 3 of Shintani’s original Dirichlet series with \(Gl_ 2({\mathbb{Z}})\)-class- numbers of integral binary cubic forms as coefficients. Residue formulas are obtained for these Dirichlet series which will be used in a future paper to compute the density of discriminants of cubic extensions of such a global field. In addition, the authors prove an identity, new even for Shintani’s series, which completely defines the relation between the distribution of \(Gl_ 2\)-classes of integral binary cubic forms and that of extensions of degree not greater than three of the base global field.

In the last section, the authors synthesize these local results with the global theory published earlier to derive the basic properties of the generalization to arbitrary global fields of characteristic not 2 or 3 of Shintani’s original Dirichlet series with \(Gl_ 2({\mathbb{Z}})\)-class- numbers of integral binary cubic forms as coefficients. Residue formulas are obtained for these Dirichlet series which will be used in a future paper to compute the density of discriminants of cubic extensions of such a global field. In addition, the authors prove an identity, new even for Shintani’s series, which completely defines the relation between the distribution of \(Gl_ 2\)-classes of integral binary cubic forms and that of extensions of degree not greater than three of the base global field.

##### MSC:

11E45 | Analytic theory (Epstein zeta functions; relations with automorphic forms and functions) |

11R56 | Adèle rings and groups |

11E76 | Forms of degree higher than two |

11E08 | Quadratic forms over local rings and fields |

11S40 | Zeta functions and \(L\)-functions |