Let be a totally real extension and an Hilbert modular cusp form of level , with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime of of residual characteristic . Let be a quadratic totally imaginary extension and be the -anticyclotomic -extension of . 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 to divides the -adic -function attached to , thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given.
Cite this article
Matteo Longo, Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms. Comment. Math. Helv. 87 (2012), no. 2, pp. 303–353DOI 10.4171/CMH/255