Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. II
type: Item
instance of: scholarly article
title: Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. II
zbMATH Open document ID: 1169.11021
published in: Publications Mathématiques
publication date: 2009-03-12T00:00:00Z
review text: Building on the methods developed by \textit{L. Clozel, M. Harris} and the author [Publ. Math., Inst. Hautes Étud. Sci. 108, 1--181 (2008; Zbl 1169.11020)], along with the ideas from the work of \textit{M. Kisin} [Moduli of finite flat group schemes and modularity, Ann. Math. (2) 170, No. 3, 1085--1180 (2009; Zbl 1201.14034)], the author proves the modularity of certain lifts of Galois representations which satisfy a list of hypotheses. Most notably, a certain ``minimality'' condition which is assumed in the paper cited above can be removed, bypassing the need of Ihara's lemma for unitary groups. Thanks to the results of the paper under review, the Sato-Tate conjecture follows (combining with the results of \textit{M. Harris, N. Shepherd-Barron} and the author [Ann. Math. (2) 171, No. 2, 779--813 (2010; Zbl 1263.11061)]) for elliptic curves over a totally real field with somewhere multiplicative reduction. We copy-paste a sample modularity theorem that the author provides in his introduction (which is Corollary 5.5 in the main body of the article): Theorem: Let \(n \in \mathbb{Z}^+\) be even and suppose \(\ell > \max\{3,n\}\) is a prime. Let \[ r: \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\longrightarrow \mathrm{GSp}_n(\mathbb{Z}_\ell) \] be a continuous irreducible representation which satisfies the following hypotheses: (\(H_R\)) \(r\) ramifies at only finitely many primes. (\(H_{C+HT}\)) \(r\big{|}_{\text{Gal}(\overline{\mathbb{Q}}_\ell/\mathbb{Q}_\ell)}\) is crystalline and \(\dim_{\mathbb{Q}_\ell} (\text{gr}^i(r\otimes B_{dR})^{\text{Gal}(\overline{\mathbb{Q}}_\ell)/\mathbb{Q}_\ell)})=0\) unless \(i \in \{0,1,\dots, n-1\}\), in which case it has dimension 1. (\(H_S\)) There is a prime \(q\neq \ell\) such that the semi-simplification \(r\big{|}_{G_{\mathbb{Q}_\ell}}^{\text{ss}}\) is unramified and \(r\big{|}_{G_{\mathbb{Q}_\ell}}^{\text{ss}}(\text{Frob}_\ell)\) has eigenvalues \(\{\alpha q^i: i=0,1, \dots, n-1\}\) for some \(\alpha\). (\(H_I\)) The image of \(…
reviewed by: Kâzım Büyükboduk
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11R34
zbMATH DE Number: 5528894
zbMATH Keywords: modularity of lifts of Galois representations
zbMATH Keywords: minimality condition
zbMATH Keywords: Sato-Tate conjecture
MaRDI profile type: Publication
full work available at URL: s10240-008-0015-2
OpenAlex ID: W2161444209
Wikidata QID: Q55952310
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DOI: 10.1007/S10240-008-0015-2
author: Richard Taylor
description: scientific article; zbMATH DE number 5528894