Overconvergent Hilbert modular forms

We generalize the construction of the eigencurve by Coleman-Mazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms. The key point is to show the overconvergence of the canonical subgroup and the complete continuity of the U_p operator. We deduce this form some general considerations in rigid analytic geometry.