Représentations galoisiennes associées aux représentations automorphes autoduales de \(GL(n)\). (Galois representations associated with self-dual automorphic representations of \(GL(n)\))
type: Item
instance of: scholarly article
title: Représentations galoisiennes associées aux représentations automorphes autoduales de \(GL(n)\). (Galois representations associated with self-dual automorphic representations of \(GL(n)\))
zbMATH Open document ID: 0739.11020
DOI: 10.1007/BF02699257
author: Laurent Clozel
published in: Publications Mathématiques
publication date: 1992-06-26T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/104076
review text: To certain automorphic representations of \(GL(n,A_ F)\), \(F\) a totally real number field, \(\ell\)-adic Galois representations are attached relating eigenvalues of Hecke operators to eigenvalues of Frobenius. This is done by first transporting the representation of \(GL(n,A_ F)\) to an appropriate unitary group \(U\). This involves base change from \(GL(n)\) over \(F\) to \(GL(n)\) over a totally imaginary quadratic extension \(F_ c\) of \(F\), correspondence between the latter and the multiplicative group \(D^*\) of a division algebra over \(F\) with involution of the second kind and descent from \(D^*\) to the unitary group \(U\). Then Kottwitz' results are applied. Essential is the hypothesis that the original representation \(\pi\) of \(GL(n,A_ F)\) be self-dual. The result implies the Ramanujan conjecture at almost all places for \(\pi\). There is an analogous result for a \(CM\)-field.
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 22762
zbMATH Keywords: automorphic representations
zbMATH Keywords: totally real number field
zbMATH Keywords: \(\ell\)-adic Galois representations
zbMATH Keywords: unitary group
zbMATH Keywords: base change
zbMATH Keywords: Ramanujan conjecture
zbMATH Keywords: \(CM\)-field
reviewed by: J. G. M. Mars
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 22762