# Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms

### Matteo Longo

Università di Padova, Italy

## 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.

## Cite this article

Matteo Longo, Anticyclotomic Iwasawa’s Main Conjecture for Hilbert modular forms. Comment. Math. Helv. 87 (2012), no. 2, pp. 303–353

DOI 10.4171/CMH/255