# 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)

DOI: http://dx.doi.org/10.1090/memo/0819

MathSciNet review: 2110225

MSC: Primary 11F41; Secondary 11F33, 11F85

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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 â€“
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 â€śĂ la Serreâ€ť as $p$-adic
uniform limit of classical modular forms, and compare them with
$p$-adic modular forms â€śĂ la Katzâ€ť 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.

Readership

Graduate students and research mathematicians interested in number
theory.

### Table of Contents

**Chapters**

- 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 Serreâ€™s $p$-adic modular forms
- 11. Katzâ€™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