On Artin's Conjecture for Odd 2-dimensional Representations
The main topic of the volume is to develop efficient algorithms by which one can verify Artin's conjecture for odd two-dimensional representations in a fairly wide range. To do this, one has to determine the number of all representations with given Artin
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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G. Frey (Ed.)
On Artin's Conjecture for Odd 2-dimensional Representations
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Authors Jacques Basmaji Ian Kirning Martin Kinzelbach Xiangdong Wang Institut fur Experimentelle Mathematik Universitat GH Essen Ellernstr. 29 D-45326 Essen, Germany Loic Merel UFR 920, Mathematiques Universite Pierre et Marie Curie 4, Place Jussieu F-75252 Paris Cedex 05
Editor Gerhard Frey Institut fur Experimentelle Mathematik Universitat GH Essen Ellernstr. 29 D-45326 Essen, Germany
Mathematics Subject Classification (1991): llFll, llR42, llF70, II Y35
ISBN 3-540-58387-4 Springer-Verlag Berlin Heidelberg New York CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany
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Foreword The investigation of the relation between the arithmetic of Galois representations and the analytic behaviour of corresponding L-series is one of the central topics in current number theory and arithmetic geometry. One of the outstanding problems in this field of research is Artin's conjecture which predicts the holomorphy of the Artin L-series of non-trivial irreducible complex representations of the absolute Galois group of number fields. Till today this conjecture is not known to be true in general; in fact even in the case of two-dimensional representations one has only partial results due to work of Heeke, Langlands and 'Iunnell, From this it follows that if r.p is an irreducible odd two-dimensional representation of the Galois group of Q then Artin's conjecture is true if the projective image of r.p is either a dihedral group (Heeke) or At (Langlands) or S... (Tunnell]. The only remaining possibility is that this image is equal to A 6 , and in this case the answer to Artin's conjecture is not known. The main concern of the following work is to develop and to implement algorithms by which one can verify Artin's conjecture for odd two-dimensional representations in a fairly wide range; it relies on considerations of J. Buhler contained in his book: "Icosahedral Galois Extensions" where he solves the problem for representations with Artin conductor 800, and in fact the attempt to systematize his work was one of the motivations of the project. Beyond th
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