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Building on our previous work on rigid analytic uniformizations, we introduceDarmon points on Jacobians of Shimura curves attached to quaternion algebrasover Q and formulate conjectures about their rationality properties. Moreover,if K is a real quadratic field, E is an elliptic curve over Q without complexmultiplication and \chi is a ring class character such that L(E/K,\chi,1) isnot 0 we prove a Gross-Zagier type formula relating Darmon points to a suitablydefined algebraic part of L(E/K,\chi,1); this generalizes results of Bertolini,Darmon and Dasgupta to the case of division quaternion algebras and arbitrarycharacters. Finally, as an application of this formula, assuming therationality conjectures for Darmon points we obtain vanishing results forSelmer groups of E over extensions of K contained in narrow ring class fieldswhen the analytic rank of E is zero, as predicted by the Birch andSwinnerton-Dyer conjecture.

Special values of L-functions and the arithmetic of Darmon points