The content of this book is taken from my manuscripts ‘On the global theory of Shintani zeta functions I–V’ which were originally intended for publication in ordinary journals. However, because of its length and the lack of a book on prehomogeneous vector spaces, it has been suggested to publish them together in book form.

It has been more than 25 years since the theory of prehomogeneous vector spaces began. Much work has been done on both the global theory and the local theory of zeta functions. However, we concentrate on the global theory in this book. I feel that another book should be written on the local theory of zeta functions in the future.

The purpose of this book is to introduce an approach based on geometric invariant theory to the global theory of zeta functions for prehomogeneous vector spaces.

This book consists of four parts. In Part I, we introduce a general formulation based on geometric invariant theory to the global theory of zeta functions for prehomogeneous vector spaces. In Part II, we apply the methods in Part I and determine the principal part of the zeta function for Siegel's case, i.e. the space of quadratic forms. In Part III, we handle relatively easy cases which are required to handle the case in Part IV. In Part IV, we use the results in Parts I–III to determine the principal part of the zeta function for the space of pairs of ternary quadratic forms.