Differential modular forms on Shimura curves over totally real fields

We study the theory of differential modular forms for compact Shimura curves over totally real fields and construct differential modular forms, which are generalizations of the fundamental differential modular forms. We also construct the Serre–Tate expansions of such differential modular forms as a possible alternative to the Fourier expansion maps and calculate the Serre–Tate expansions of some of these differential modular forms.