Potential automorphy and change of weight
type: Item
instance of: scholarly article
title: Potential automorphy and change of weight
zbMATH Open document ID: 1310.11060
author: Thomas Barnet-Lamb
author: Toby Gee
author: David Geraghty
author: Richard Taylor
published in: Annals of Mathematics. Second Series
publication date: 2014-04-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1010.2561
review text: The automorphy of Galois representations, as part of the Langlands program, is one of the most important problems in number theory. The present paper studies the potential automorphy of weakly compatible systems of \(l\)-adic representations of the absolute Galois group of a totally real number field. More precisely, let \(F\) be a number field, and \(G_F\) its absolute Galois group. A family of continuous semi-simple representations \[ r_\lambda : G_F\to\mathrm{GL}_n(\overline{M}_\lambda), \] indexed by finite places \(\lambda\) of a number field \(M\), is a weakly compatible system of \(n\)-dimensional \(l\)-adic Galois representations unramified outside a finite set \(S\) of places of \(F\) provided -- \(r_\lambda\) is unramified at all finite places \(v\not \in S\) for \(\lambda\) not dividing the residual characteristic of \(v\), and the characteristic polynomial of \(r_\lambda(\mathrm{Frob}_v)\) is a polynomial over \(M\) independent of \(\lambda\); -- \(r_\lambda\) is de Rham at all places \(v\) of \(F\) lying above the residual characteristic of \(\lambda\) and crystalline at all \(v\not\in S\) with that property; -- the Hodge-Tate numbers of \(r_\lambda\) are independent of \(\lambda\) for each embedding of \(F\) into the algebraic closure of \(M\). The main result of the paper is that a weakly compatible system, as defined above, is potentially automorphic if it is irreducible (all \(r_\lambda\) are irreducible), regular (every \(r_\lambda\) has \(n\) different Hodge-Tate numbers for any embedding of \(F\)), and odd essentially self-dual (\(F\) is totally real, \(r_\lambda\) all factor either through symplectic or orthogonal group with a parity condition on the multiplier character, and the multiplier characters form a weakly compatible system). For simplicity it is also assumed that the image of each embedding of \(F\) into the algebraic closure of \(M\) is actually contained in \(M\). Potential automorphy means that there is a finite Galois totally real …
reviewed by: Neven Grbac
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 22E50
Mathematics Subject Classification ID: 22E55
zbMATH DE Number: 6284344
zbMATH Keywords: automorphic forms
zbMATH Keywords: Galois representations
zbMATH Keywords: weakly compatibly system
zbMATH Keywords: \(l\)-adic representations
zbMATH Keywords: potential automorphy
zbMATH Keywords: automorphy lifting
MaRDI profile type: Publication
OpenAlex ID: W2123527372
arXiv ID: 1010.2561
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Potential automorphy and change of weight (DOI)
DOI: 10.4007/ANNALS.2014.179.2.3