Contents. 1. Introduction 1. 2. Notations 5. 3. Moduli spaces of abelian varieties with real multiplication 6. 4. Properties of S 9. 5. Hilbert modular forms 15. 6. The ^-expansion map 16. 7. The partial Hasse invariants 21. 8. Reduceness of the partial Hasse invariants 28. 9. A compactification of Wl(k, fipN)Kum 36. 10. Congruences mod pn and Serre's p-adic modular forms 39. 11. Katz's p-adic Hilbert modular forms 43. 12. The operators Oy,i 50. 13. The operator V 67. 14. The operator U 72. 15. Applications to filtrations of modular forms 75. 16. Theta cycles and parallel filtration (inert case) 80. 17. Functorialities 84. 18. Integrality and congruences for values of zeta functions 88. 19. Numerical examples 94. 20. Comments regarding values of zeta functions 96. 21. References 98. v
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