On the Functional Equations Satisfied by Eisenstein Series
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544 Robert P. Langlands
On the Functional Equations Satisfied by Eisenstein Series
Springer-Verlag Berlin. Heidelberg 9New York 19?6
Author Robert P. Langlands School of Mathematics Institute for Advanced Study Princeton, N.J. 08540/USA
Library of Congress Cataloging in Publication Data
Langlands, Robert P 1936On the functional equations satisfied by Eisenstein series.
(Lecture notes in mathematics ; 544) Bibliography: p. Includes index. i. Eisenstein series. 2. Functional equations. 3. Lie groups. 4. Automorphlc forms. I. Title. II. Series: Lecture notes in mathematics (Berlin) ;
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QA3.L?.8 no. 544 [@,A}~04] 510'.8s [515%7]
76-41389
AMS Subject Classifications (1970): 10C15, 10C99, 22E40, 30A58, 32 N10, 42A16, 43A65 ISBN 3-540-07872-X Springer-Verlag Berlin 9 Heidelberg 9 New York ISBN 0-387-07872-X Springer-Verlag New York 9 Heidelberg 9 Berlin This .work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under w 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. 9 by Springer-Verlag Berlin 9 Heidelberg 1976 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr.
PREFACE In t h e s e d a y s of d i z z y i n g s c i e n t i f i c p r o g r e s s w h e n o f f e r i n g to t h e m a t h e m a t i c a l certainly bears
s o m e apology i s c a l l e d f o r
public a work written twelve years
t h e s t a m p of a j u v e n i l e h a n d ,
it c o m p l e t e l y ,
trace
and the manuscript
There were two possibilities:
o r to p u b l i s h it a s it s t o o d ; a n d I p r e f e r r e d
There were, cation.
about my neck.
w h e n it w a s f i r s t w r i t t e n ,
T h e s t u d y of E i s e n s t e i n
formula,
series
and the t r a c e f o r m u l a
It
a n d I h a d a l w a y s h o p e d to r e v i s e
b u t m y i n c l i n a t i o n to a r e a l e f f o r t g r e w e v e r s l i g h t e r , c o m i n g an a l b a t r o s s
ago.
was be-
to f o r g e t a b o u t
the second.
other reasons
is a prelim ina ry
it,
for delaying publi-
to t h e d e v e l o p m e n t
has been a long time evolving.
of a
Not only does
it present serious analytic difficulties, but also the uses to w h i c h it should be put have not b e e n clear.
A sustained attack on the analytic difficulties is n o w being
carried out, by A r t h u r and others, and, thanks to a large extent to d e v e l o p m e n t s within the theory of Eisenstein series itself, w e n o w have a clearer picture of the theorems he done,
that will flow f r o m the trace formula.
However
and a complete treatment of Eisenstein series,
a great deal r e m a i n s to even imperfect,
may
be
useful to those wishing to try their hand at developing or using t
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