Arthur's Invariant Trace Formula and Comparison of Inner Forms
This monograph provides an accessible and comprehensive introduction to James Arthur’s invariant trace formula, a crucial tool in the theory of automorphic representations. It synthesizes two decades of Arthur’s research and writing into one volume,
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Arthur’s Invariant Trace Formula and Comparison of Inner Forms
Yuval Z. Flicker
Arthur’s Invariant Trace Formula and Comparison of Inner Forms
Yuval Z. Flicker Ariel University Ariel, Israel The Ohio State University Columbus, Ohio, USA
ISBN 978-3-319-31591-1 DOI 10.1007/978-3-319-31593-5
ISBN 978-3-319-31593-5 (eBook)
Library of Congress Control Number: 2016940094 Mathematics Subject Classification (2010): 11F70, 11F72, 11F41, 11M36, 22E55, 22E57 © Springer International Publishing Switzerland 2016 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Birkhäuser imprint is published by Springer Nature The registered company is Springer International Publishing AG, CH
Preface
The theory of automorphic representations of the group G.A/ of the adèle points of a reductive connected group G over a global field F, and that of admissible representations of the group G.Fv / of points of a reductive connected group G over a local field Fv , are governed by a hypothetical reciprocity law, introduced by Langlands, that relates them to representations of a variant of the Galois group of the base field, named Weil or Weil-Deligne group, into the complex Langlands dual group L G of G. This “principle of functoriality”—not touched upon in the present tome— suggests relations between such automorphic and admissible representations of different groups G. These relations have been termed liftings, correspondences, transfers, and are suggested by relations amongst the underlying dual groups. For example, establishing lifting from GL.2/ to GL.n C 1/ corresponding to the irreducible n-dimensional representation Symn from the dual group GL.2; C/ to GL.n C 1; C/ would imply the Ramanujan conjecture for GL.2/. Some of these liftings, which are analytic implications of the principle, have been established by various techniques, using various invariants of the representations. The work of Jacquet-Langlands [JL70] showed that the Selberg trace formula [Se62] could
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