On the p-adic uniformization of quaternionic Shimura curves

Abstract Let D be a quaternion algebra over a totally real number field F that splits exactly at one infinite place of F . The associated Shimura curve $${\textrm{Sh}}$$ Sh is canonically defined over F . Let $$\mathfrak {p}_0 | p$$ p 0 | p be a prime of F such that $$D_{\mathfrak {p}_0}$$ D p 0 is a division algebra. Let $$\textbf{K} \subset (D \otimes \mathbb {A}_f)^{\times }$$ K ⊂ ( D ⊗ A f ) × be an open compcact subgroup such that $$\textbf{K}_{\mathfrak {p}_0}$$ K p 0 is maximal. $${\textrm{Sh}}_{\textbf{K}}$$ Sh K has a unique stable model over $$O_{F_{\mathfrak {p}_0}}$$ O F p 0 . The p -adic unifomization describes completion of this model by quotients of the p -adic upper halfplane $$\hat{\Omega }_{F_{\mathfrak {p}_0}}^2$$ Ω ^ F p 0 2 by actions of discrete subgroups. It was discovered by Cherednik. Our result generalizes a more precise theorem of Drinfeld in the case $$F = \mathbb {Q}$$ F = Q .