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Arithmeticity in the Theory of Automorphic Forms

Goro Shimura Princeton University, Princeton, NJ
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Softcover ISBN: 978-0-8218-4961-3
Product Code: SURV/82.S
302 pp
List Price: $90.00 MAA Member Price:$81.00
AMS Member Price: $72.00 Electronic ISBN: 978-1-4704-1309-5 Product Code: SURV/82.S.E 302 pp List Price:$84.00
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List Price: $135.00 MAA Member Price:$121.50
AMS Member Price: $108.00 Click above image for expanded view Arithmeticity in the Theory of Automorphic Forms Goro Shimura Princeton University, Princeton, NJ Available Formats:  Softcover ISBN: 978-0-8218-4961-3 Product Code: SURV/82.S 302 pp  List Price:$90.00 MAA Member Price: $81.00 AMS Member Price:$72.00
 Electronic ISBN: 978-1-4704-1309-5 Product Code: SURV/82.S.E 302 pp
 List Price: $84.00 MAA Member Price:$75.60 AMS Member Price: $67.20 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version. List Price:$135.00
MAA Member Price: $121.50 AMS Member Price:$108.00
• Book Details

Mathematical Surveys and Monographs
Volume: 822000
MSC: Primary 11; 14; 32;

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.

• Chapters
• I. Automorphic forms and families of abelian varieties
• II. Arithmeticity of automorphic forms
• III. Arithmetic of differential operators and nearly holomorphic functions
• IV. Eisenstein series of simpler types
• V. Zeta functions associated with Hecke eigenforms
• VI. Analytic continuation and near holomorphy of Eisenstein series of general types
• VII. Arithmeticity of the critical values of zeta functions and Eisenstein series of general types

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Volume: 822000
MSC: Primary 11; 14; 32;

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.

• Chapters
• I. Automorphic forms and families of abelian varieties
• II. Arithmeticity of automorphic forms
• III. Arithmetic of differential operators and nearly holomorphic functions
• IV. Eisenstein series of simpler types
• V. Zeta functions associated with Hecke eigenforms
• VI. Analytic continuation and near holomorphy of Eisenstein series of general types
• VII. Arithmeticity of the critical values of zeta functions and Eisenstein series of general types
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