Zeta functions of prehomogeneous affine spaces

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

— In this paper we consider the prehomogeneous vector space for a pair of simple algebras which are inner forms of the D4 type and the E6 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 when the simple algebras are non-split. We also give a description of the sets of rational ...

On the Zeta Functions of Prehomogeneous Vector Spaces for Pair of Simple Algebras

In this paper, we consider the prehomogeneous vector space for pair of simple algebras which are k-forms of the D4 type and the E6 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,...

COMPUTATION OF CHARACTER SUMS AND APPLICATIONS TO PREHOMOGENEOUS VECTOR SPACES 1 with an appendix ” L - FUNCTIONS OF PREHOMOGENEOUS VECTOR SPACES

For an arbitrary number field K with ring of integers OK , we prove an analogue over finite rings of the form OK/p of the Fundamental Theorem on the Fourier transform of a relative invariant of prehomogeneous vector spaces, where p is a big enough prime ideal of OK and m > 1. In the appendix, F. Sato gives an application of the Theorems 1.2 and 1.5 (and Theorems A, B, C in [4]) to the functiona...

Geometric zeta-functions of locally symmetric spaces

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes the Patterson conjecture. We also extend the range of zeta functions in con...

Prehomogeneous Vector Spaces and Field Extensions Ii