  Building on our previous work on rigid analytic uniformizations, we introduce
Darmon points on Jacobians of Shimura curves attached to quaternion algebras
over Q and formulate conjectures about their rationality properties. Moreover,
if K is a real quadratic field, E is an elliptic curve over Q without complex
multiplication and \chi is a ring class character such that L(E/K,\chi,1) is
not 0 we prove a Gross-Zagier type formula relating Darmon points to a suitably
defined algebraic part of L(E/K,\chi,1); this generalizes results of Bertolini,
Darmon and Dasgupta to the case of division quaternion algebras and arbitrary
characters. Finally, as an application of this formula, assuming the
rationality conjectures for Darmon points we obtain vanishing results for
Selmer groups of E over extensions of K contained in narrow ring class fields
when the analytic rank of E is zero, as predicted by the Birch and
Swinnerton-Dyer conjecture.
