TABLE OF CONTENTS Preface vii Notation and Terminology ix Introduction 1 Chapter I. Automorphic Forms and Families of Abelian Varieties 7 1. Algebraic preliminaries 7 2. Polarized abelian varieties 13 3. Symmetric spaces and factors of automorphy 17 4. Families of polarized abelian varieties 22 5. Definition of automorphic forms 30 6. Parametrization by theta functions 37 Chapter II. Arithmeticity of Automorphic Forms 45 7. The field Xo(Qab) _ 45 8. Action of certain elements of G A on A 50 9. The reciprocity-law at CM-points and rationality of auto- morphic forms 58 10. Automorphisms of the spaces of automorphic forms 67 11. Arithmeticity at CM-points 76 Chapter III. Arithmetic of Differential Operators and Nearly Holomorphic Functions 87 12. Differential operators on symmetric spaces 87 13. Nearly holomorphic functions 98 14. Arithmeticity of nearly holomorphic functions 107 15. Holomorphic projection 122 Chapter IV. Eisenstein Series of Simpler Types 127 16. Eisenstein series on U(rjn) 127 17. Arithmeticity and near holomorphy of Eisenstein series 137 18. Eisenstein series in the Hilbert modular case 149 Chapter V. Zeta Functions Associated with Hecke Eigenforms 159 19. Formal Euler products and generalized Mobius functions 159 20. Dirichlet series obtained from Hecke eigenvalues and Fourier coefficients 166 21. The Euler products for the forms of half-integral weight 175 22. The largest possible pole of Z(s, f, x) 177

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