On the p-adic Bloch-Kato conjecture for Hilbert modular forms

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. A generalization of this conjecture to all geometric Galois representations V was formulated by Bloch and Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when V 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 on E(Q) constructed by Heegner. The proof in the general case relies on the universal p-adic deformation of Heegner points and a formula for its height.