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We study Hilbert modular forms in characteristic p and over padic 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 Θχ and study the variation of the filtration under these operators. In particular, we prove that every ordinary eigenform has filtration in a prescribed box of weights. Our methods are geometriccomparing 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 the 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 Θχ to the p-adic setting.

Let M be the Shimura variety associated with the group of spinor similitudes of a quadratic space over Q of signature (n, 2). We prove a conjecture of Bruinier-Kudla-Yang, relating the arithmetic intersection multiplicities of special divisors and big CM points on M to the central derivatives of certain L-functions.As an application of this result, we prove an averaged version of Colmez's conjecture on the Faltings heights of CM abelian varieties.

We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication.

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