# Hilbert Modular Forms: mod \(p\) and \(p\)-Adic Aspects

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*F. Andreatta; E. Z. Goren*

We study Hilbert modular forms in characteristic \(p\) and
over \(p\)-adic rings. In the characteristic \(p\) theory we
describe the kernel and image of the \(q\)-expansion map and prove the
existence of filtration for Hilbert modular forms; we define operators
\(U\), \(V\) and \(\Theta_\chi\) and study the variation
of the filtration under these operators. Our methods are geometric –
comparing holomorphic Hilbert modular forms with rational functions on a moduli
scheme with level-\(p\) structure, whose poles are supported on the
non-ordinary locus.

In the \(p\)-adic theory we study congruences between Hilbert modular
forms. This applies to the study of congruences between special values of zeta
functions of totally real fields. It also allows us to define \(p\)-adic
Hilbert modular forms “à la Serre” as \(p\)-adic
uniform limit of classical modular forms, and compare them with
\(p\)-adic modular forms “à la Katz” that are regular
functions on a certain formal moduli scheme. We show that the two notions agree
for cusp forms and for a suitable class of weights containing all the classical
ones. We extend the operators \(V\) and \(\Theta_\chi\) to the
\(p\)-adic setting.

#### Table of Contents

# Table of Contents

## Hilbert Modular Forms: mod $p$ and $p$-Adic Aspects

- Contents v6 free
- 1. Introduction 18 free
- 2. Notations 512 free
- 3. Moduli spaces of abelian varieties with real multiplication 613
- 4. Properties of G 916
- 5. Hilbert modular forms 1522
- 6. The q-expansion map 1623
- 7. The partial Hasse invariants 2128
- 8. Reduceness of the partial Hasse invariants 2835
- 9. A compactification of m(k, μ[sub(pN)])[sup(Kum)] 3643
- 10. Congruences mod p[sup(n)] and Serre's p-adic modular forms 3946
- 11. Katz's p-adic Hilbert modular forms 4350
- 12. The operators Θ[sub(B,i)] 5057
- 13. The operator V 6774
- 14. The operator U 7279
- 15. Applications to filtrations of modular forms 7582
- 16. Theta cycles and parallel filtration (inert case) 8087
- 17. Functorialities 8491
- 18. Integrality and congruences for values of zeta functions 8895
- 19. Numerical examples 94101
- 20. Comments regarding values of zeta functions 96103
- 21. References 98105