عددی بودن در نظریه اشکال اتومورفیک (بررسی‌ها و مونوگرافی‌های ریاضی)

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. Automorphic Forms And Families Of Abelian Varieties -- Algebraic Preliminaries -- Polarized Abelian Varieties -- Symmetric Spaces And Factors Of Automorphy -- Families Of Polarized Abelian Varieties -- Definition Of Automorphic Forms -- Parametrization By Theta Functions -- Arithmeticity Of Automorphic Forms -- The Field A[subscript 0](q[subscript Ab]) -- Action Of Certain Elements Of G[subscript A] On [characters Not Reproducible] -- The Reciprocity-law At Cm-points And Rationality Of Automorphic Forms -- Automorphisms Of The Spaces Of Automorphic Forms -- Arithmeticity At Cm-points -- Arithmetic Of Differential Operators And Nearly Holomorphic Functions -- Differential Operators On Symmetric Spaces -- Nearly Holomorphic Functions -- Arithmeticity Of Nearly Holomorphic Functions -- Holomorphic Projection -- Eisenstein Series Of Simpler Types -- Eisenstein Series On U ([eta Subscript N]) -- Arithmeticity And Near Holomorphy Of Eisenstein Series -- Eisenstein Series In The Hilbert Modular Case -- Zeta Functions Associated With Hecke Eigenforms -- Formal Euler Products And Generalized Mobius Functions -- Dirichlet Series Obtained From Hecke Eigenvalues And Fourier Coefficients -- The Euler Products For The Forms Of Half-integral Weight -- The Largest Possible Pole Of Z (s, F, X) -- Analytic Continuation And Near Holomorphy Of Eisenstein Series Of General Types -- Eisenstein Series Of General Types -- Pullback Of Eisenstein Series -- Proof Of Theorems In Sections 20 And 23 -- Near Holomorphy Of Eisenstein Series In Case Ub. Goro Shimura. Includes Bibliographical References (p. 297-299) And Index. Describes Eisenstein series and zeta functions that are associated with Hecke eigenforms on symplectic and unitary groups. This book discusses many basic problems that arise in number-theoretical investigations of automorphic forms but that cannot be found in expository forms. It is suitable for researchers in the field of zeta functions.