The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. Generalisations of this conjecture to motives M were formulated by Belinson and Bloch-Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when M is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1.
The case of elliptic curves corresponds to classical modular forms of weight two, and was treated by Perrin-Riou in 1987 using the modular points in E(Q) constructed by Heegner. The proof in the general case is based on the universal p-adic deformation of Heegner points, via a formula for its height.