About this Title
Hiroyuki Yoshida, Kyoto University, Kyoto, Japan
Publication: Mathematical Surveys and Monographs
Publication Year 2003: Volume 106
ISBNs: 978-0-8218-3453-4 (print); 978-1-4704-1333-0 (online)
MathSciNet review: MR2011848
MSC: Primary 11G15; Secondary 11F67, 11F70, 11J89, 11M41, 11R33, 11R42
The central theme of this book is an invariant attached to an ideal class of a totally real algebraic number field. This invariant provides us a unified understanding of periods of abelian varieties with complex multiplication and the Stark-Shintani units. This is a new point of view, and the book contains many new results related to it.
To place these in proper perspective and to supply tools to attack unsolved problems, the author gives systematic expositions of fundamental topics. Thus the book treats the multiple gamma function, the Stark conjecture, Shimura's period symbol, the absolute period symbol, Eisenstein series on $GL(2)$, and a limit formula of Kronecker's type. The discussion of each of these topics is enhanced by many illustrative examples. The major part of the text is written assuming, in addition to basic knowledge, some familiarity with algebraic number theory. About thirty problems are included for exercises, some of which are quite challenging.
The book is intended for graduate students and researchers working in number theory and automorphic forms.
Graduate students and research mathematicians interested in number theory and algebraic geometry.
Table of Contents
- I. Multiple gamma function and its generalizations
- II. The Stark–Shintani conjecture
- III. Absolute CM–periods
- IV. Explicit cone decompositions and applications
- V. Applications of a limit formula of Kronecker’s type
- Appendix I. Eisenstein series on $GL(2)$
- Appendix II. On higher derivatives of $L$-functions
- Appendix III. Transcendental property of CM-periods