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

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 is a ring class character such that its complex L-function over K does not vanish, we prove a Gross–Zagier type formula relating Darmon points to a suitably defined algebraic part of the L-function; 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.