Automorphic Forms on GL(2) Part II
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278 Herve Jacquet The City University of New York, New York, NYfUSA
Automorphic Forms on GL(2) Part II
Springer-Verlag Berlin· Heidelberg· NewYork 1972
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
278 Herve Jacquet The City University of New York, New York, NYfUSA
Automorphic Forms on GL(2) Part II
Springer-Verlag Berlin· Heidelberg· NewYork 1972
AMS Subject Classifications (1970): lOD15
ISBN 3-540-05931-8 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05931-8 Springer-Verlag New York· Heidelberg· 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 § 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.
© by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 76-108338. Printed in Germany. Offsetdruck: Julius Beltz, Hemshach/Bergstr.
Introduction This is a continuation of "Automorphic Forms on
GL (2) ".
Un-
fortunately, the reader (if any) will have to have a serious knowledge of the two first chapters of the first volume if he is to find his way through the second one.
Perhaps reading Godement's
on Jacquet-
Langlands", Institute for Advanced Study (1970) will help him in satisfying this stringent requirement.
The main purpose of the second
volume is to reformulate and extend a classical result:
if
are two Dirichlet series associated with automorphic forms (in the classical sense) then the Dirichlet series \' a bins n n
L
is convergent in some right half space, can be analytically continued in the whole complex plane as a meromorphic function of a suitable functional equation.
s
and satisfies
Anything novel in this work comes from
the point of view which is the theory of group representations.
The
local theory in §14 to 18 is a preparation of a technical nature for the global theory of § 19. the latter section.
The motivations appear therefore only in
The reader should read first §14, take for granted
the results of §16 to §18 and then go to §19. to quadratic extensions.
§20 is an application
Again there is nothing really new in it.
In the Bibliography I have tried to indicate my indebtness to previous authors.
But I could not however acknowledge completely my
indebtness to G. Shimura.
This paper would have never been written
IV if not for a suggestion of his.
In particular, the application to
quadratic extensions of §20 was, after the oral indications he gave to me, a routine exercise. I gratefully acknowledge the support of The City University of New York and the National Science Foundation (GP 27952). to express
my
I wish also
thanks to Mrs. Sophie Gerber for typing these not
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