Anticyclotomic Iwasawa's Main Conjecture for Hilbert modular forms

Abstract

Let F/ℚ beatotally real extension and f an Hilbertmodular cusp formof level n, with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime S | n of F of residual characteristic p. Let K/F be a quadratic totally imaginary extension and K S∞ be the S-anticyclotomic ℤ p-extension of K. The main result of this paper, generalizing the analogous result [5] 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 (f, K S∞) divides the p-adic L-function attached to (f, K S∞), thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given. © Swiss Mathematical Society.