The works of Katz-Mazur and Deligne-Rapoport provide a description of an integral model of classical modular curves like X(N), X_0(N) and various variants of X_1(N). In particular, if p is a prime dividing N, this model is not smooth over Z_p and we know how the special fiber looks like. It is a union of irreducible components meeting at the supersingular points, in a way that can be made precise, heavily depending on r=ord_p(N).

I have been looking for similar results in the literature for Shimura curves associated to quaternion algebras over totally real number fields, and I found only partial results. I am aware of the classical paper of Carayol and the more recent papers of Frazer Jarvis and Toby Gee, but there are many cases that remain to be covered in these references, specially when F is not Q and r=ord_p(N) is >1. Here p is a prime ideal and N is an integral ideal in F.

The difficulty of course lies in the fact that when F is not the field of rational numbers, the moduli interpretation is more tricky. In that case these curves are best viewed as boundary components of an unitary Shimura variety.

My question is: do you know of more recent works where this problem is settled?


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