On the zeta functions of prehomogeneous vector spaces for pair of simple algebras

Abstract

In this paper, we consider the prehomogeneous vector space for pair of simple algebras which are $k$-forms of the $D_4$ type and the $E_6$ type. We mainly study the non-split cases. The main purpose of this paper is to determine the principal parts of the global zeta functions associated with these spaces in the non-split cases. We also give a description of the sets of rational orbits of these spaces, which suggests the expected density theorems arising from the properties of these zeta functions.