Memoirs of the American Mathematical Society 2005; 100 pp; softcover Volume: 173 ISBN10: 0821836099 ISBN13: 9780821836095 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 nonordinary 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. Readership Graduate students and research mathematicians interested in number theory. Table of Contents  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
