Arithmeticity in the theory of automorphic forms / Goro Shimura.

Format:

Book

Author/Creator:

Shimura, Gorō, 1930- author.

Series:

Mathematical surveys and monographs ; no. 82.

Mathematical surveys and monographs, 0076-5376 ; volume 82

Language:

English

Subjects (All):

Automorphic forms.

Physical Description:

1 online resource (314 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [2000]

Summary:

Written by one of the leading experts, venerable grandmasters, and most active contributors $\ldots$ in the arithmetic theory of automorphic forms $\ldots$ the new material included here is mainly the outcome of his extensive work $\ldots$ over the last eight years $\ldots$ a very careful, detailed introduction to the subject $\ldots$ this monograph is an important, comprehensively written and profound treatise on some recent achievements in the theory. --Zentralblatt MATH The main objects of study in this book are Eisenstein series and zeta functions associated with Hecke eigenforms on symplectic and unitary groups. After preliminaries--including a section, ``Notation and Terminology''--the first part of the book deals with automorphic forms on such groups. In particular, their rationality over a number field is defined and discussed in connection with the group action; also the reciprocity law for the values of automorphic functions at CM-points is proved. Next, certain differential operators that raise the weight are investigated in higher dimension. The notion of nearly holomorphic functions is introduced, and their arithmeticity is defined. As applications of these, the arithmeticity of the critical values of zeta functions and Eisenstein series is proved. Though the arithmeticity is given as the ultimate main result, the book discusses many basic problems that arise in number-theoretical investigations of automorphic forms but that cannot be found in expository forms. Examples of this include the space of automorphic forms spanned by cusp forms and certain Eisenstein series, transformation formulas of theta series, estimate of the Fourier coefficients of modular forms, and modular forms of half-integral weight. All these are treated in higher-dimensional cases. The volume concludes with an Appendix and an Index. The book will be of interest to graduate students and researchers in the field of zeta functions and modular forms.

Contents:

""Table of Contents""; ""Preface""; ""Notation and Terminology""; ""Introduction""; ""Chapter I. Automorphic Forms and Families of Abelian Varieties""; ""1. Algebraic preliminaries""; ""2. Polarized abelian varieties""; ""3. Symmetric spaces and factors of automorphy""; ""4. Families of polarized abelian varieties""; ""5. Definition of automorphic forms""; ""6. Parametrization by theta functions""; ""Chapter II. Arithmeticity of Automorphic Forms""; ""7. The field A[sub(0)](Q[sub(ab)])""; ""8. Action of certain elements of G[sub(A)] on R""

""9. The reciprocity-law at CM-points and rationality of automorphic forms""""10. Automorphisms of the spaces of automorphic forms""; ""11. Arithmeticity at CM-points""; ""Chapter III. Arithmetic of Differential Operators and Nearly Holomorphic Functions""; ""12. Differential operators on symmetric spaces""; ""13. Nearly holomorphic functions""; ""14. Arithmeticity of nearly holomorphic functions""; ""15. Holomorphic projection""; ""Chapter IV. Eisenstein Series of Simpler Types""; ""16. Eisenstein series on U(η[sub(n)])""; ""17. Arithmeticity and near holomorphy of Eisenstein series""

""25. Proof of Theorems in Sections 20 and 23""""26. Near holomorphy of Eisenstein series in Case UB""; ""Chapter VII. Arithmeticity of the Critical Values of Zeta Functions and Eisenstein Series of General Types""; ""27. The spaces of holomorphic Eisenstein series""; ""28. Main theorems on arithmeticity in Cases SP and UT""; ""29. Main theorems on arithmeticity in Case UB""; ""Appendix""; ""A1. The series associated to a symmetric matrix and Gauss sums""; ""A2. Metaplectic groups and factors of automorphy""; ""A3. Transformation formulas of general theta series""

""A4. The constant term of a theta series at each cusp depends only on the genus""""A5. Theta series of a hermitian form""; ""A6. Estimate of the Fourier coefficients of a modular form""; ""A7. The Mellin transforms of Hilbert modular forms""; ""A8. Certain unitarizable representation spaces""; ""References""; ""Index""; ""A""; ""C""; ""D""; ""F""; ""G""; ""H""; ""I""; ""L""; ""M""; ""N""; ""P""; ""Q""; ""R""; ""S""; ""U""

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

1-4704-1309-4