Shintani Zeta Functions

This is amongst the first books on the theory of prehomogeneous vector spaces, and represents the author's deep study of the subject.

Akihiko Yukie (Author)

9780521448048, Cambridge University Press

Paperback, published 3 February 1994

352 pages
22.8 x 15.2 x 2 cm, 0.507 kg

The theory of prehomogeneous vector spaces is a relatively new subject although its origin can be traced back through the works of Siegel to Gauss. The study of the zeta functions related to prehomogeneous vector spaces can yield interesting information on the asymptotic properties of associated objects, such as field extensions and ideal classes. This is amongst the first books on this topic, and represents the author's deep study of prehomogeneous vector spaces. Here the author's aim is to generalise Shintani's approach from the viewpoint of geometric invariant theory, and in some special cases he also determines not only the pole structure but also the principal part of the zeta function. This book will be of great interest to all serious workers in analytic number theory.

Introduction
1. The general theory
2. Eisenstein series
3. The general program
4. The zeta function for the spaces
5. The case G=GL(2)¥GL(2), V=Sym2 k2ƒk2
6. The case G=GL(2)¥GL(1)2, V=Sym2 k2ƒk
7. The case G=GL(2)¥GL(1), V=Sym2 k2ƒk2
8. Invariant theory of pairs of ternary quadratic forms
9. Preliminary estimates
10. The non-constant terms associated with unstable strata
11. Unstable distributions
12. Contributions from unstable strata
13. The main theorem.

Subject Areas: Calculus & mathematical analysis [PBK]