Anticyclotomic Iwasawa's main conjecture for Hilbert modular forms
type: Item
instance of: scholarly article
title: Anticyclotomic Iwasawa's main conjecture for Hilbert modular forms
zbMATH Open document ID: 1306.11087
DOI: 10.4171/CMH/255
author: Matteo Longo
published in: Commentarii Mathematici Helvetici
publication date: 2012-06-05T00:00:00Z
review text: Let \(F\) be a totally real number field and \(f\) a 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 \(\wp\) of \(F\) dividing \(\mathfrak{n}\), of residual characteristic \(p \leq 5\). Let \(K\) be a quadratic imaginary extension of \(F\) and \(K_{\wp^\infty}\) be the \(\wp\)-anticyclotomic \(\mathbb{Z}_p\)-extension of \(K\). Suppose for simplicity that \(p\) is unramified in \(F\) and in \(K_f\), where \(K_f\) is the field generated over \(\mathbb Q\) by the eigenvalues of the Hecke algebra acting on \(f\). Fix an embedding of \(\overline{\mathbb Q}\) into \(\overline{\mathbb Q}_{p}\) and denote by \(\pi\) the prime ideal of the ring of integers \(O_f\) of \(K_f\). Assuming certain technical conditions on \(\wp\) and \(\mathfrak{n}\), it is possible to associate to f a \(p\)-adic \(L\)-function \(L_{\wp,\pi} (f/K)\) which is an element of the Iwasawa algebra \(\Lambda_{\wp,\pi}\) of the Galois group of \(K_{\wp^\infty }/K\) over the completion \(O_{f,\pi}\) of \(O_f\) at \(\pi\). The so-called anticyclotomic Iwasawa Main Conjecture (AMC for short) states that the ideals of \(\Lambda_{\wp,\pi}\) generated by \(L_{\wp,\pi} (f/K)\) and by the characteristic series \([[\text{Char}]]_{\wp,\pi} (f/K)\) of the Pontryagin dual of the Selmer group attached to \((f, K_{\wp^\infty})\) must coincide. The central result of this paper is that, under suitable arithmetic assumptions and some technical conditions (among them, a condition which consists in a version of Ihara's lemma for Shimura curves over totally real fields), \([[\text{Char}]]_{\wp,\pi} (f/K)\) divides \(L_{\wp,\pi} (f/K)\). This generalizes the analogous result proved by \textit{M. Bertolini} and \textit{H. Darmon} when \(F = \mathbb Q\) and \(O_{f,\pi} = \mathbb Z_p\) [Ann. Math. (2) 162, No. 1, 1--64 (2005; Zbl 1093.11037)]. The author hopes that, using the techniques announced by …
reviewed by: Thong Nguyen Quang Do
Mathematics Subject Classification ID: 11R23
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11G10
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 6043061
zbMATH Keywords: anticyclotomic Iwasawa main conjecture
zbMATH Keywords: Hilbert modular forms
MaRDI profile type: Publication
Wikidata QID: Q122962276
full work available at URL: https://doi.org/10.4171/cmh/255
OpenAlex ID: W2079571699
description: scientific article; zbMATH DE number 6043061