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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