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 parameterizes the weights. Finally, we define Hecke operators. We focus on the $\mathrm{U}$ operator, showing that it is completely continuous on the space of overconvergent modular forms.
Proposition 2.1.2 is false: it is never used in the rest of the thesis, so skipping it it is not a problem. Section 4.1 is slightly wrong, in particular the description of the weight space given at the end of page 40 is false. Also Lemma 4.1.7 needs to be changed. See the paper (Section 6.1) for a corrected version. In any case all that matters is Remark 4.1.8, and, if one uses the correct $\mathcal{W}_r$, this is true (the difference is really small and quite technical, so you can read the thesis unless you are really interested in all the details).