Periods and points attached to quadratic algebras

Φ : H/Γ0(N) −→ E(C), (0–1) where H is the Poincaré upper half-plane and Γ0(N) is Hecke’s congruence group of level N . Fix a quadratic field K; when it is imaginary, the theory of complex multiplication combined with (0–1) yields the construction of a remarkable collection of points on E defined over certain ring class fields of K. These are the Heegner points recalled in Section 1 whose study is one of the themes of this proceedings volume. When p is a prime that divides N exactly, and E a factor of the Jacobian of a Shimura curve attached to a quaternion algebra ramified at p, the uniformisation Φ of (0–1) admits a p-adic analogue based on theorems of Jacquet–Langlands and Cerednik–Drinfeld, in which H is replaced by Drinfeld’s p-adic upper half plane Hp := Cp − Qp, and Γ0(N) by an appropriate {p}-arithmetic subgroup