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Hilbert modular forms: mod $p$ and $p$-adic aspects

About this Title

F. Andreatta and E. Z. Goren

Publication: Memoirs of the American Mathematical Society
Publication Year 2005: Volume 173, Number 819
ISBNs: 978-0-8218-3609-5 (print); 978-1-4704-0420-8 (online)
MathSciNet review: 2110225
MSC: Primary 11F41; Secondary 11F33, 11F85

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Table of Contents


  • 1. Introduction
  • 2. Notations
  • 3. Moduli spaces of abelian varieties with real multiplication
  • 4. Properties of $\mathcal {G}$
  • 5. Hilbert modular forms
  • 6. The $q$-expansion map
  • 7. The partial Hasse invariants
  • 8. Reduceness of the partial Hasse invariants
  • 9. A compactification of $\mathfrak {M}(k, \mu _{pN})^{Kum}$
  • 10. Congruences mod $p^n$ and Serre’s $p$-adic modular forms
  • 11. Katz’s $p$-adic Hilbert modular forms
  • 12. The operators $\Theta _{\mathfrak {P},i}$
  • 13. The operator $V$
  • 14. The operator $U$
  • 15. Applications to filtrations of modular forms
  • 16. Theta cycles and parallel filtration (inert case)
  • 17. Functorialities
  • 18. Integrality and congruences for values of zeta functions
  • 19. Numerical examples
  • 20. Comments regarding values of zeta functions