Abstract

This paper presents the D module approach for the “geometric” study of the solutions and “generalized” solutions of a system of differential operators with holomorphic coefficients. One associates to such a system the quotient ring D/I of the ring D by a left ideal I and studies the properties of the complexes Rhom(D/I, F) (see subsection 1.4).

If we have a finite representation of the coefficients, for instance when they are polynomials, many results developed below are effective. Also many constructions which look rather abstract can be mechanized, in particular through standard basis computations (see [B.M.] [C] [G]). However the complexity of these algorithms is too high and makes them intractable in practice. The result of [Gr] shows that the membership problem has a double exponential complexity. A more geometric theory with simple exponential complexity (like in commutative algebra) does not exist yet and should be developed …

The first part of the paper presents in detail several constructions in the one variable case. The second part provides an introduction to holonomic D modules and Bernstein polynomials, it relies on the paper of B. Malgrange “Motivations and introduction to the theory of D module” which is published in this volume. We give an algorithm and a new bound for the computation of Bernstein's polynomial in the case of an isolated singularity.

This chapter and the next chapter are the technical heart of this book. In this chapter, we estimate the Eisenstein series on GL(n) for the Borel subgroup B. This reduces to the estimate of Whittaker functions on GL(n). The estimate of the Whittaker functions consists of two problems. One is at finite places, and the other is at infinite places. We prove the explicit formula for the p-adic Whittaker functions in §2.3 following Shintani [66]. Even though some estimate of Whittaker functions are known, the uniformity of the estimate is a big issue in this book, because we intend to consider contour integrals. Therefore, it is necessary to prove an estimate which is of polynomial growth with respect to the parameter of the Lie algebra. For this purpose, we generalize Shintani's approach in [64], which is the naive use of integration by parts. As far as GL(3) is concerned, Bump's estimate [5] is the best possible estimate. Our estimate, even though it is enough for our purposes for the moment, may not be the optimum estimate, and we may have to prove a better estimate in the future to handle more prehomogeneous vector spaces.

The Fourier expansion of automorphic forms on GL(n)

In this section, we review the notion of the Fourier expansion of automorphic functions on GL(n) following Piatecki–Shapiro in [51].

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.