Eisenstein Series and Applications
Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. Bringing together con
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Series Editors Hyman Bass Joseph Oesterl´e Alan Weinstein
Eisenstein Series and Applications Wee Teck Gan Stephen S. Kudla Yuri Tschinkel Editors
Birkh¨auser Boston • Basel • Berlin
Stephen S. Kudla Department of Mathematics University of Toronto 40 St. George Street Toronto, Ontario M5S 2E4 Canada [email protected]
Wee Teck Gan Department of Mathematics University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093 U.S.A. [email protected] Yuri Tschinkel Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 10012 U.S.A. [email protected]
ISBN-13: 978-0-8176-4496-3 DOI: 10.1007/978-0-8176-4639-4
e-ISBN-13: 978-0-8176–4639-4
Library of Congress Control Number: 2007937323 Mathematics Subject Classification (2000): 11F70, 22E55, 11F67, 32N15 c 2008 Birkh¨auser Boston All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media LLC, 233 Spring Street, New York, NY 10013, USA) and the author, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
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To Robert Langlands, on the occasion of his seventieth birthday
Preface
The theory of Eisenstein series, in the general form given to it by Robert Langlands some forty years ago, has been an important and incredibly useful tool in the fields of automorphic forms, representation theory, number theory and arithmetic geometry. For example, the theory of automorphic L-functions arises out of the calculation of the constant terms of Eisenstein series along parabolic subgroups. Not surprisingly, the two primary approaches to the analytic properties of automorphic L-functions, namely the Langlands–Shahidi method and the Rankin–Selberg method, both rely on the theory of Eisenstein series. In representation theory, Eisenstein series were originally studied by Langlands in order to give the spectral decomposition of the space of L2 functions of locally symmetric spaces attached to adelic groups. This spectral theory has been used to prove the unitarity of certain local representations. Finally, on the more arithmetic side, the Fourier coefficients of Eisenstein series contain a wealth of arithmetic information which is far from being completely understood. The p-divisibility properties of these coefficients, for example, are instrumental in the construction of p-adic L-functions. In short, the theory of Eisenstein series seems to have, hidden within it, an inexhaustible number of treasures waiting to be
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