Shintani zeta functions / Akihiko Yukie.

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

pt. I. The general theory. Ch. 1. Preliminaries. Ch. 2. Eisenstein series on GL(n). Ch. 3. The general program -- pt. II. The Siegel-Shintani case. Ch. 4. The zeta function for the space of quadratic forms -- pt. III. Preliminaries for the quartic case. Ch. 5. The case G = GL(2) x GL(2), V = Sym[superscript 2]k[superscript 2] [actual symbol not reproducible] k[superscript 2]. Ch. 6. The case G = GL(2) x GL(1)[superscript 2], V = Sym[superscript 2]k[superscript 2] [actual symbol not reproducible] k. Ch. 7. The case G = GL(2) x GL(1)[superscript 2], V = Sym[superscript 2]k[superscript 2] [actual symbol not reproducible] k[superscript 2] -- pt. IV. The quartic case. Ch. 8. Invariant theory of pairs of ternary quadratic forms. Ch. 9. Preliminary estimates. Ch. 10. The non-constant terms associated with unstable strata. Ch. 11. Unstable distributions. Ch. 12. Contributions front unstable strata. Ch. 13. The main theorem.

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.