Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms

Abstract

Let $F/\mathbb{Q}$ be a totally real extension and $f$ an Hilbert modular cusp form of level $\mathfrak{n}$, with trivial central character and parallel weight 2, which is an eigenform for the action of the Hecke algebra. Fix a prime $\wp | \mathfrak{n}$ of $F$ of residual characteristic $p$. Let $K/F$ be a quadratic totally imaginary extension and $K\_{\wp^\infty}$ be the $\wp$-anticyclotomic $\mathbb{Z}p$-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 $(f,K{\wp^\infty})$ divides the $p$-adic $L$-function attached to $(f,K\_{\wp^\infty})$, thus proving one direction of the Anticyclotomic Main Conjecture for Hilbert modular forms. Arithmetic applications are given.