On a conjecture of Ash
type: Item
instance of: scholarly article
title: On a conjecture of Ash
zbMATH Open document ID: 1007.11020
author: Adrian Barbu
published in: Journal of Algebra
publication date: 2002-09-08T00:00:00Z
review text: The author shows that for certain level-\(p\)-subgroups \(\Gamma\) of GL\((n,{\mathbb Z})\) the Hecke eigenclasses in \(H^*(\Gamma,{\mathbb F}_p)\) which are induced from \(H^*(\Gamma\bmod p,{\mathbb F}_p)\) have Hecke polynomials (for almost every prime \(\ell\)) which coincide with det\((I-\rho(\text{Frob}_\ell) X)\) for some \(n\)-dimensional \(\mathbb{F}_p G\) module, where \(G\) is the absolute Galois group of the rationals and Frob\(_\ell\) some choice of Frobenius element at \(\ell.\) The representation is constructed as a sum of powers of the cyclotomic character. The result confirms a conjecture of \textit{A. Ash} [Duke Math. J. 65, 235-255 (1992; Zbl 0774.11024)] in a particular case.
reviewed by: Stefan Kühnlein
Mathematics Subject Classification ID: 11E72
Mathematics Subject Classification ID: 11F80
zbMATH DE Number: 1797468
zbMATH Keywords: Hecke pair
zbMATH Keywords: cohomology
zbMATH Keywords: Galois representation
zbMATH Keywords: cyclotomic character
MaRDI profile type: Publication
OpenAlex ID: W1967775377
Wikidata QID: Q122914838
cites work: Galois representations attached to mod \(p\) cohomology of \(GL(n,\mathbb{Z})\)
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cites work: Hecke operators in cohomology of groups
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DOI: 10.1006/JABR.2001.9140
description: scientific article; zbMATH DE number 1797468