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Hilbert Modular Forms: mod $$p$$ and $$p$$-Adic Aspects
F. Andreatta, University Degli Studi, Padova, Italy, and E. Z. Goren, McGill University, Montreal, QC, Canada
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Memoirs of the American Mathematical Society
2005; 100 pp; softcover
Volume: 173
ISBN-10: 0-8218-3609-9
ISBN-13: 978-0-8218-3609-5
List Price: US$66 Individual Members: US$39.60
Institutional Members: US\$52.80
Order Code: MEMO/173/819

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.

Graduate students and research mathematicians interested in number theory.

• Introduction
• Notations
• Moduli spaces of abelian varieties with real multiplication
• Properties of $$\mathcal{G}$$
• Hilbert modular forms
• The $$q$$-expansion map
• The partial Hasse invariants
• Reduceness of the partial Hasse invariants
• A compactification of $$\mathfrak{M} (k,\mu_{pN})^{\rm{Kum}}$$
• Congruences mod $$p^n$$ and Serre's $$p$$-adic modular forms
• Katz's $$p$$-adic Hilbert modular forms
• The operators $$\Theta_{\mathfrak{P},i}$$
• The operator $$V$$
• The operator $$U$$
• Applications to filtrations of modular forms
• Theta cycles and parallel filtration (inert case)
• Functorialities
• Integrality and congruences for values of zeta functions
• Numerical examples
• Comments regarding values of zeta functions
• References