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.