Eisenstein series are an essential ingredient in the spectral theory of automorphic forms and an important tool in the theory of <EM>L-functions. They have also been exploited extensively by number theorists for many arithmetic purposes. Bringing together contributions from areas that are not usually interacting with each other, this volume introduces diverse users of Eisenstein series to a variety of important applications. With this juxtaposition of perspectives, the reader obtains deeper insights into the arithmetic of Eisenstein series.
The exposition focuses on the common structural properties of Eisenstein series occurring in many related applications that have arisen in several recent developments in arithmetic: Arakelov intersection theory on Shimura varieties, special values of <EM>L-functions and Iwasawa theory, and equidistribution of rational/integer points on homogeneous varieties. Key questions that are considered include: Is it possible to identify a class of Eisenstein series whose Fourier coefficients (resp. special values) encode significant arithmetic information? Do such series fit into p-adic families? Are the Eisenstein series that arise in counting problems of this type?
Contributors include: B. Brubaker, D. Bump, J. Franke, S. Friedberg, W.T. Gan, P. Garrett, M. Harris, D. Jiang, S.S. Kudla, E. Lapid, K. Prasanna, A. Raghuram, F. Shahidi, R. Takloo-Bighash
Progress in Mathematics Volume 258
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 t