Weights in Serre's conjecture for \(\text{GL}_n\) via the Bernstein-Gelfand-Gelfand complex
type: Item
instance of: scholarly article
title: Weights in Serre's conjecture for \(\text{GL}_n\) via the Bernstein-Gelfand-Gelfand complex
zbMATH Open document ID: 1222.11070
author: Michael M. Schein
published in: Journal of Number Theory
publication date: 2011-04-20T00:00:00Z
review text: The nice article under review is concerned with the weights in an \(n\)-dimensional generalization of Serre's Modularity Conjecture [\textit{J.-P. Serre}, Duke Math. J. 54, 179--230 (1987; Zbl 0641.10026)]. More precisely, its aim is to determine for any \textit{modular} Galois representation \(\rho: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_n(\overline{\mathbb{F}}_p)\) of a \(p\)-minute weight (see below) which Fontaine-Lafaille numbers can at most occur for the restriction of \(\rho\) to a decomposition group at~\(p\). The notions of modularity and Serre weights adopted by the author need explanation. Differing from the notion of modularity in Serre type conjectures by Ash, Doud, Pollack, Sinnott [\textit{A. Ash, D. Doud} and \textit{D. Pollack}, Duke Math. J. 112, No. 3, 521--579 (2002; Zbl 1023.11025); \textit{A. Ash} and \textit{W. Sinnott}, Duke Math. J. 105, No. 1, 1--24 (2000; Zbl 1015.11018)] and \textit{F. Herzig} [``The weight in a Serre-type conjecture for tame \(n\)-dimensional Galois representations'', Duke Math. J. 149, No. 1, 37--116 (2009; Zbl 1232.11065)], the author calls a \(\rho\) as above modular if it occurs in the étale cohomology of a certain compact Shimura variety defined over an imaginary quadratic field in which \(p\) splits. A Serre weight is defined as an isomorphism class of an irreducible \(\overline{\mathbb{F}}_p[\mathrm{GL}_n(\mathbb{F}_p)]\)-module, and \(\rho\) is modular of Serre weight~\(V\) if it occurs in the étale cohomology with the locally constant étale sheaf associated with~\(V\). The Serre weights are parametrized by so-called \(p\)-restricted tuples \((a_1,\dots,a_n)\) (by definition: \(a_1 \geq a_2 \geq \dots \geq a_n\) and \(0 \leq a_i - a_{i+1} \leq p-1\) for \(1 \leq i \leq n-1\)) such that \(0 \leq a_n < p-1\). The weight is called \(p\)-minute if the attached \(p\)-restricted tuple satisfies in addition \(a_1-a_n < p-n\) and \(a_1+a_2+\dots+a_{n-1} - (n-1)a_n < p-1\). The main theorem of the …
reviewed by: Gabor Wiese
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 14F30
zbMATH DE Number: 5880813
zbMATH Keywords: Serre's conjecture
zbMATH Keywords: Galois representations
zbMATH Keywords: crystalline cohomology
MaRDI profile type: Publication
full work available at URL: j.jnt.2008.03.008
OpenAlex ID: W2036603240
Wikidata QID: Q122949689
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DOI: 10.1016/J.JNT.2008.03.008
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description: scientific article; zbMATH DE number 5880813