Hardcover ISBN:  9780821834534 
Product Code:  SURV/106 
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AMS Member Price:  $78.40 
Electronic ISBN:  9781470413330 
Product Code:  SURV/106.E 
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Book DetailsMathematical Surveys and MonographsVolume: 106; 2003; 282 ppMSC: Primary 11; 33; Secondary 30;
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 StarkShintani 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.ReadershipGraduate students and research mathematicians interested in number theory and algebraic geometry.

Table of Contents

Chapters

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 CMperiods


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

Chapters

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 CMperiods