Anticycotomic Iwasawa's Main Conjecture for Hilbert modular forms

Let F be a totally real extension and f an Hilbert modular cusp form of level n, with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime P of F of residual characteristic p. Let K be a quadratic totally imaginary extension of F and K' be the P-anticyclotomic Zp-extension of K. The main result of this paper, generalizing the analogous result of Bertolini and Darmon, states that, under suitable arithmetic assumptions and some technical restrictions, the characteristic power series of the Pontryagin dual of the Selmer group attached of f over K' divides the p-adic L-function attached to f and K' thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given.