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