Absolute CM-periods / Hiroyuki Yoshida.

Format:

Book

Author/Creator:

Yoshida, Hiroyuki, 1947- author.

Series:

Mathematical surveys and monographs ; no. 106.

Mathematical surveys and monographs, 0076-5376 ; volume 106

Language:

English

Subjects (All):

Abelian varieties.

Automorphic forms.

Functions, Zeta.

Physical Description:

1 online resource (296 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2003]

Language Note:

English

Summary:

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 results in proper perspective and to supply tools to attack unsolved problems, the author gives systematic expositions offundamental 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 textis 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.

Contents:

""Table of Contents""; ""Preface""; ""Notation and Terminology""; ""Introduction""; ""Chapter I. MULTIPLE GAMMA FUNCTION AND ITS GENERALIZATIONS""; ""Â1. Basic integral representations""; ""Â2. Shintani's formulas""; ""Â3. The second derivative of ς(s,A,x) at s = 0""; ""Â4. An asymptotic expansion of the multiple gamma function""; ""Exercises""; ""Chapter II. THE STARKâ€?SHINTANI CONJECTURE""; ""Â1. Stark's general conjecture""; ""Â2. Transition to a more precise conjecture""; ""Â3. Shintani's formulas for the partial zeta function""; ""Â4. An example""; ""Exercises""

""Chapter III. ABSOLUTE CMâ€?PERIODS""""Â1. Shimura's period symbol PK""; ""Â2. The case of abelian fields""; ""Â3. Conjectures on absolute CMâ€?periods""; ""Â4. Numerical examples""; ""Â5. Further investigations on the invariant X(c)""; ""Â6. Numerical examples (continued)""; ""Exercises""; ""Chapter IV. EXPLICIT CONE DECOMPOSITIONS AND APPLICATIONS""; ""Â1. A special decomposition of a higher dimensional cube""; ""Â2. Topological preparations""; ""Â3. A sufficient condition for a cone decomposition""; ""Â4. Examples""

""Â5. Explicit cone decompositions for index finite subgroups""""Â6. Applications""; ""Exercises""; ""Chapter V. APPLICATIONS OF A LIMIT FORMULA OF KRONECKER'S TYPE""; ""Â1. A limit formula of Kronecker's type""; ""Â2. A generalization of the exact Chowlaâ€?Selberg formula""; ""Â3. L-functions of orders of an algebraic number field""; ""Â4. Toward the reciprocity law for the h-function""; ""Â5. A connection of automorphic forms with group cohomology""; ""Exercises""; ""Appendix I. EISENSTEIN SERIES ON GL(2)""; ""Â1. Eisenstein series on GL(2)""

""Â2. Calculations of local integrals""""Â3. The functional equation of Eisenstein series""; ""Â4. Eisenstein series of class 1""; ""Appendix II. ON HIGHER DERIVATIVES OF L-FUNCTIONS""; ""Â1. A search for new invariants""; ""Â2. The case of the second derivatives of Lâ€?functions""; ""Appendix III. TRANSCENDENTAL PROPERTY OF CMâ€?PERIODS""; ""Exercises""; ""References""; ""Index""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 275-279) and index.

Description based on print version record.

ISBN:

1-4704-1333-7