# Is there a concrete way to show the existence of canonical model for non-modular Shimura curves?

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!

• Hi Nicolas, welcome to MO. I think too many things in this question are referred to in an vague way. Try adding links to these papers, and maybe add a definition or two which will make it easier to read. – Amir Sagiv Jun 2 '17 at 13:19
• Hi Thanks, I added a couple of references for context. Adding definitions in the question would not be practical as they are pretty involved. – Nicolás Jun 2 '17 at 13:26
• Nicolas, is it important for the sake of Carayol's argument that the model used be canonical, in Deligne's sense or in whatever sense this word may have? I am not familiar with the details, but I wouldn't be surprised if much less was needed of the model. Then the question of being canonical will fade away, and it should be easy to construct a model with the required property, for instance as the solution of a suitable moduli problem. – Joël Jun 2 '17 at 15:22
• Hi Joël, thanks for the comment. To be honest I am not sure how important the "canonical" part is, but I think the real issue is that the varieties considered are known not to 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. – Nicolás Jun 2 '17 at 15:27
• One "less modular" approach is to have an imbedding of a given group into a symplectic group (with its more usual models), known (for some specific reason) to hit "many" CM-points or other special subvarieties. Then rationality facts from the larger model can be used to infer those on the imbedded variety... Some aspects of what Deligne did were of this sort. – paul garrett Jun 2 '17 at 17:28