Galois representations attached to Hilbert-Siegel modular forms
type: Item
instance of: scholarly article
title: Galois representations attached to Hilbert-Siegel modular forms
zbMATH Open document ID: 1246.11114
published in: Documenta Mathematica
publication date: 2010-12-15T00:00:00Z
full work available at URL: 19.html
review text: The article under review concerns the construction and properties of Galois representations attached to Hilbert-Siegel modular forms of genus~\(2\). More precisely, the main theorem of the article reunites three important results. Let \(F\) be a totally real number field and \(\pi\) a globally generic cuspidal automorphic representation of \(\mathrm{GSp}(4)\) such that \(\pi_\infty\) is regular algebraic and such that \(\pi\) has a Steinberg component at some finite place. (a) For a fixed embedding \(\iota: \overline{\mathbb{Q}}_\ell \hookrightarrow \mathbb{C}\), the author establishes the existence of a Galois representation \[ \rho_{\pi,\iota}: \mathrm{Gal}(\overline{F}/F) \to \mathrm{GSp}_4(\overline{\mathbb{Q}}_\ell) \] which satisfies local-global compatibility at all places \(v \nmid \ell\). Moreover, local information is provided for the restriction of \(\rho_{\pi,\iota}\) to places above~\(\ell\). We note that the local Langlands correspondence for \(\mathrm{GSp}(4)\) was established by \textit{W. T. Gan} and \textit{S. Takeda} [``The local Langlands conjecture for \(\mathrm{GSp}(4)\)'', Ann. Math. (2) 173, No. 3, 1841--1882 (2011; Zbl 1230.11063)]. This result generalises and extends work of \textit{G. Laumon} [``Fonctions zêtas des variétés de Siegel de dimension trois'', Astérisque 302, 1--66 (2005; Zbl 1097.11021)], \textit{R. Taylor} [``On the \(l\)-adic cohomology of Siegel threefolds'', Invent. Math. 114, No. 2, 289--310 (1993; Zbl 0810.11034)], and \textit{R. Weissauer} [``Four dimensional Galois representations'', Astérisque 302, 67--150 (2005; Zbl 1097.11027)] for the case of \(F=\mathbb{Q}\). The main ingredients in the construction are the fundamental work by Harris and Taylor establishing the local Langlands correspondence for \(\mathrm{GL}(n)\) [\textit{M. Harris} and \textit{R. Taylor}, The geometry and cohomology of some simple Shimura varieties. Princeton, NJ: Princeton University Press (2001; Zbl 1036.11027)] and the subsequent work of Tayl…
reviewed by: Gabor Wiese
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11F41
Mathematics Subject Classification ID: 11F70
Mathematics Subject Classification ID: 11F46
zbMATH DE Number: 5827287
zbMATH Keywords: Galois representations
zbMATH Keywords: Hilbert-Siegel modular forms
zbMATH Keywords: monodromy
author: Claus M. Sorensen
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5827287