**Mathematical Surveys and Monographs**

Volume: 106;
2003;
282 pp;
Hardcover

MSC: Primary 11; 33;
Secondary 30

Print ISBN: 978-0-8218-3453-4

Product Code: SURV/106

List Price: $98.00

AMS Member Price: $78.40

MAA Member Price: $88.20

**Electronic ISBN: 978-1-4704-1333-0
Product Code: SURV/106.E**

List Price: $92.00

AMS Member Price: $73.60

MAA Member Price: $82.80

# Absolute CM-Periods

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*Hiroyuki Yoshida*

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 with 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 results 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 examples. The majority of the text is
written assuming some familiarity with algebraic number theory. About thirty
problems are included, some of which are quite challenging.

The book is intended for graduate students and researchers working in number
theory and automorphic forms.

#### Readership

Graduate students and research mathematicians interested in number theory and algebraic geometry.

#### Table of Contents

# Table of Contents

## Absolute CM-Periods

- Table of Contents v6 free
- Preface vii8 free
- Notation and Terminology ix10 free
- Introduction 112 free
- Chapter I. MULTIPLE GAMMA FUNCTION AND ITS GENERALIZATIONS 1324 free
- Chapter II. THE STARK–SHINTANI CONJECTURE 4354
- Chapter III. ABSOLUTE CM–PERIODS 6172
- Chapter IV. EXPLICIT CONE DECOMPOSITIONS AND APPLICATIONS 135146
- Chapter V. APPLICATIONS OF A LIMIT FORMULA OF KRONECKER'S TYPE 169180
- §1. A limit formula of Kronecker's type 169180
- §2. A generalization of the exact Chowla–Selberg formula 179190
- §3. L-functions of orders of an algebraic number field 189200
- §4. Toward the reciprocity law for the h-function 197208
- §5. A connection of automorphic forms with group cohomology 207218
- Exercises 213224

- Appendix I. EISENSTEIN SERIES ON GL(2) 215226
- Appendix II. ON HIGHER DERIVATIVES OF L-FUNCTIONS 249260
- Appendix III. TRANSCENDENTAL PROPERTY OF CM–PERIODS 269280
- References 275286
- Index 281292 free