Icosahedral Galois extensions and elliptic curves
type: Item
instance of: scholarly article
title: Icosahedral Galois extensions and elliptic curves
zbMATH Open document ID: 0885.11036
DOI: 10.1007/BF02677473
author: Annette Klute
published in: Manuscripta Mathematica
publication date: 1998-04-22T00:00:00Z
full work available at URL: https://eudml.org/doc/156301
review text: This paper is devoted to the last unsolved case of the Artin conjecture in two dimensions. Given an irreducible two-dimensional complex representation of the absolute Galois group of a number field \(F\), the Artin conjecture states that the associated \(L\)-function is entire. The conjecture has been proven in all cases except the icosahedral one. The way one proves it is to use lifting techniques to construct a cuspidal modular form whose \(L\)-function coincides with the given one. In the paper, icosahedral representations \(\rho\) for \(F=\mathbb{Q}(\sqrt 5)\) are constructed and their \(L\)-series are computed explicitly. Then it is shown that the existence of a modular form \(f\) whose \(L\)-function coincides with \(L(s,\rho)\) only up to a certain ideal will already imply the Artin conjecture for this representation \(\rho\).
Mathematics Subject Classification ID: 11F66
Mathematics Subject Classification ID: 11G05
zbMATH DE Number: 1038293
zbMATH Keywords: icosahedral Galois extensions
zbMATH Keywords: elliptic curves
zbMATH Keywords: Artin conjecture
zbMATH Keywords: absolute Galois group
zbMATH Keywords: \(L\)-function
zbMATH Keywords: icosahedral representation
reviewed by: Anton Deitmar
describes a project that uses: ecdata
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1038293