  In this work, we set up a theory of p-adic modular forms over Shimura curves
over totally real fields which allows us to consider also non-integral weights.
In particular, we define an analogue of the sheaves of k-th invariant
differentials over the Shimura curves we are interested in, for any p-adic
character. In this way, we are able to introduce the notion of overconvergent
modular form of any p-adic weight. Moreover, our sheaves can be put in p-adic
families over a suitable rigid-analytic space, that parametrizes the weights.
Finally, we define Hecke operators, including the U operator, that acts
compactly on the space of overconvergent modular forms. We also construct the
eigencurve.
