Absolute CM-Periods (Mathematical Surveys and Monographs Series)

Absolute CM-Periods (Mathematical Surveys and Monographs Series)

by Hiroyuki Yoshida
     
 

ISBN-10: 0821834533

ISBN-13: 9780821834534

Pub. Date: 11/30/2003

Publisher: American Mathematical Society

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

Overview

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.

Product Details

ISBN-13:
9780821834534
Publisher:
American Mathematical Society
Publication date:
11/30/2003
Series:
Mathematical Surveys and Monographs Series, #106
Pages:
282
Product dimensions:
7.09(w) x 9.84(h) x (d)

Table of Contents

Prefacevii
Notation and Terminologyix
Introduction1
Chapter I.Multiple Gamma Function and Its Generalizations13
1.Basic integral representations13
2.Shintani's formulas17
3.The second derivative of [zeta](s, A, x) at s = 024
4.An asymptotic expansion of the multiple gamma function34
Exercises41
Chapter II.The Stark-Shintani Conjecture43
1.Stark's general conjecture43
2.Transition to a more precise conjecture48
3.Shintani's formulas for the partial zeta function52
4.An example54
Exercises58
Chapter III.Absolute CM-periods61
1.Shimura's period symbol pk61
2.The case of abelian fields65
3.Conjectures on absolute CM-periods74
4.Numerical examples94
5.Further investigations on the invariant X(c)108
6.Numerical examples (continued)126
Exercises133
Chapter IV.Explicit Cone Decompositions and Applications135
1.A special decomposition of a higher dimensional cube135
2.Topological preparations139
3.A sufficient condition for a cone decomposition144
4.Examples147
5.Explicit cone decompositions for index finite subgroups149
6.Applications161
Exercises166
Chapter V.Applications of a Limit Formula of Kronecker's Type169
1.A limit formula of Kronecker's type169
2.A generalization of the exact Chowla-Selberg formula179
3.L-functions of orders of an algebraic number field189
4.Toward the reciprocity law for the h-function197
5.A connection of automorphic forms with group cohomology207
Exercises213
Appendix I.Eisenstein Series on GL(2)215
1.Eisenstein series on GL(2)215
2.Calculations of local integrals222
3.The functional equation of Eisenstein series234
4.Eisenstein series of class 1243
Appendix II.On Higher Derivatives of L-Functions249
1.A search for new invariants249
2.The case of the second derivatives of L-functions253
Appendix III.Transcendental Property of CM-periods269
Exercises274
References275
Index281

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