Lectures on Shimura Curves 8: Real Points

To be more precise, let F be a totally real field of degree g, of narrow class number 1, and B/F a quaternion algebra split at exactly one infinite place ∞1 of F . Let N be an integral ideal prime to the discriminant of B. Let us write Γ, Γ0(N ), Γ1(N ), Γ(N ) for the corresponding congruence subgroups of Γ, the image in PGL2(R) of the positive norm units of a maximal order O (unique up to conjugacy, by our class number assumption) of B. We write X = Γ\H, X0(N ) = Γ0(N )\H, X1(N ) = Γ1(N )\H, X(N ) = Γ(N )\H. In fact, for brevity, let us write X•(N ) for a statement which is valid for any of the above curves.