On icosahedral Artin representations
type: Item
instance of: scholarly article
title: On icosahedral Artin representations
zbMATH Open document ID: 1015.11021
DOI: 10.1215/S0012-7094-01-10922-8
author: Kevin Buzzard
author: Mark Dickinson
author: N. I. Shepherd-Barron
author: Richard Taylor
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
review text: The authors prove the (strong) Artin conjecture in the icosahedral case for an infinite class of representations. Let \(\rho: \text{Gal} (\overline{\mathbb{Q}}/\mathbb{Q})\to \text{GL}_2 (\mathbb{C})\) be an odd, continuous, irreducible representation. They prove that for \(\rho\) icosahedral (under some restrictions) the Artin \(L\)-series for \(\rho\) is the Mellin transform of a weight 1 newform and is an entire function (Theorem A). The proof follows a strategy outlined in [\textit{R. Taylor}, Pac. J. Math., Spec. Iss., 337-347 (1998; Zbl 0942.11031)]. The main result follows from the special case (Theorem 4.1) and slight extensions of results in [\textit{N. Shepherd-Barron} and \textit{R. Taylor}, J. Am. Math. Soc. 10, 283-298 (1997; Zbl 1015.11019), \textit{M. Dickinson}, Duke Math. J. 109, 319-382 (2001; Zbl 1015.11020) and \textit{K. Buzzard} and \textit{R. Taylor}, Ann. Math. (2) 149, 905-919 (1999; Zbl 0965.11019)].
reviewed by: Andrzej Dąbrowski
Mathematics Subject Classification ID: 11F80
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11R42
zbMATH DE Number: 1820840
zbMATH Keywords: Artin representations
zbMATH Keywords: Artin \(L\)-series
zbMATH Keywords: modular form
zbMATH Keywords: modular curve
zbMATH Keywords: Hida theory
zbMATH Keywords: universal deformation
MaRDI profile type: Publication
full work available at URL: s0012-7094-01-10922-8
OpenAlex ID: W2082481329
description: scientific article; zbMATH DE number 1820840