Book Review: Arithmeticity in the theory of automorphic forms

Abstract

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.