I am trying to read Carayol's article on the construction of Galois representations associated to Hilbert modular forms (http://archive.numdam.org/article/CM_1986__59_2_151_0.pdf). The main geometric ingredient is a study of the reduction of (non-modular) Shimura curves associated to a quaternion algebra over a totally real number field $F$ over $Q$, where $F \neq Q$.

In that article the existence of a canonical model (in the sense of Deligne) is deduced from Deligne's general results about general Shimura varieties (https://publications.ias.edu/sites/default/files/34_VarietesdeShimura.pdf). If I remember correctly Deligne's proof is very general and works by showing the result for entire classes of groups $G$ at once.

I was wondering if there was an easier way to show existence in this particular case ?

Thanks!

notto be solutions to moduli problems of the usual type, so some other indirect route is needed to establish the existence of models over (some extension of) Q. $\endgroup$ – Nicolás Jun 2 '17 at 15:27