Distributions of discriminants of cubic algebras

Abstract

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 $\mathbb{Z}$.

37pages

Cited by in corpus (3)

• On relations among Dirichlet series whose coefficients are class numbers of binary cubic forms
• On parameterizations of rational orbits of some forms of prehomogeneous vector spaces
• The subregular unipotent contribution to the geometric side of the Arthur trace formula for the split exceptional group $G_2$