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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We prove a gluing lemma for sections of line bundles on a rigid analytic variety. We apply the lemma in conjunction with a result of Buzzard [Bu, Theorem 5.2] to give a proof of (a generalization of) Coleman's theorem, which states that overconvergent modular forms of small slope are classical. The proof is geometric in nature and is suitable for generalization to other Shimura varieties.Let f be an overconvergent U p -eigenform of weight k with eigenvalue a p . If the p-adic valuation of a p is less than k − 1, then f is classical.This so-called control theorem is crucial in many applications of the theory to modular forms. In many problems, one can reduce the rigidity of the situation by working in Banach spaces of overconvergent forms and yet in the end get results about classical modular forms by invoking the control theorem. For example, it is easier to construct p-adic families of overconvergent eigenforms and then identify classical

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We set up the basic theory of P-adic modular forms over certain unitary PEL Shimura curves M K . For any PEL abelian scheme classified by M K , which is not 'too supersingular', we construct a canonical subgroup which is essentially a lifting of the kernel of Frobenius from characteristic p. Using this construction we define the U and Frob operators in this context. Following Coleman, we study the spectral theory of the action of U on families of overconvergent P-adic modular forms and prove that the dimension of overconvergent eigenforms of U of a given slope is a locally constant function of the weight.