Distributions of Discriminants of Cubic Algebras
نویسنده
چکیده
We study the space of binary cubic and quadratic forms over the ring of integers O of an algebraic number field k. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta functions for these spaces. Applying these zeta functions, we derive some density theorems on the distributions of discriminants of cubic algebras of O. In the case k is a quadratic field, we give a correction term as well as the main term. These are generalizations of Shintani’s asymptotic formulae of the mean values of class numbers of binary cubic forms over Z.
منابع مشابه
Distributions of Discriminants of Cubic Algebras Ii
Let k be a number field and O the ring of integers. In the previous paper [T06] we study the Dirichlet series counting discriminants of cubic algebras of O and derive some density theorems on distributions of the discriminants by using the theory of zeta functions of prehomogeneous vector spaces. In this paper we consider these objects under imposing finite number of splitting conditions at non...
متن کاملGraduate School of Mathematical Sciences Komaba, Tokyo, Japan Distributions of Discriminants of Cubic Algebras
We study the space of binary cubic and quadratic forms over the ring of integers O of an algebraic number field k. By applying the theory of prehomogeneous vector spaces founded by M. Sato and T. Shintani, we can associate the zeta functions for these spaces. Applying these zeta functions, we derive some density theorems on the distributions of discriminants of cubic algebras of O. In the case ...
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