On the p-adic uniformization of quaternionic Shimura curves

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 Sh Sh is canonically defined over F . Let p₀ | p p 0 | p be a prime of F such that Dp₀ D p 0 is a division algebra. Let K ⊂ (D ⊗ Af)× K ⊂ ( D ⊗ A f ) × be an open compcact subgroup such that Kp₀ K p 0 is maximal. ShK Sh K has a unique stable model over O_Fp₀ O F p 0 . The p -adic unifomization describes completion of this model by quotients of the p -adic upper halfplane Ω ̂_Fp₀² 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 = Q F = Q .