Adeles and Algebraic Groups
This volume contains the original lecture notes presented by A. Weil in which the concept of adeles was first introduced, in conjunction with various aspects of C.L. Siegel’s work on quadratic forms. These notes have been supplemented by an extended bibli
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Progress in Mathematics Vol. 23 Edited by J. Coates and S. Helgason
Birkhauser
Boston· Basel· Stuttgart
A. Weil
Adeles and Algebraic Groups
1982
Birkhauser Boston • Basel • Stuttgart
Author: A. Weil The Institute for Advanced Study Princeton, New Jersey 08540
Library of Congress Cataloging in Publication Data Weil, Andr~, 1906Adeles and algebraic groups. (Progress in mathematics; v. 23) "Notes are based on lectures, given at the Institute for Advanced Study in 1959-1960"-Foreword. Bibliography: p. 1. Forms, Quadratic. 2. Linear algebraic groups. 3. Adeles. I. Title. II. Series: Progress in mathematics (Cambridge, Mass.) ; v. 23. 512.9'44 82-12767 QA243.W44 1982
CIP - Kurztitelauf der Deutschen Bibliothek (-lei 1, Andre: Adeles and alnebraic groups / Andre ~eil. Boston; Basel; Stuttqart : BirkhMuser, 1982. (Progress in mathematics; 23)
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All riqhts reserved. No part of this publication may be reoroduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without orior permission of the coPyri0ht owner. It> Bi rkhauser Boston, 1982
ISBN 978-1-4684-9158-6 ISBN 978-1-4684-9156-2 (eBook) DOI 10.1007/978-1-4684-9156-2
FaR E waR 0 The present notes are based on lectures, given at the Institute for Advanced Study in 1959-1960, which, in a sense, were nothing but a commentary on various aspects of Siegel's work-chiefly his classical papers on quadratic forms, but also the later papers where the volumes of various fundamental domains are computed. The very fruitful idea of applying the adele method to such problems comes from Tamagawa, whose work on this subject is not yet published; was able to make use of a manuscript of his, where that idea was applied to the restatement and proof of Siegel's theorem on quadratic forms. If the reader is able to derive some profit from these notes, he will owe it, to a large extent, to M. Demazure and T. ano, who have greatly improved upon the oral presentation of this material as given in my lectures. At many points they have acted as collaborators rather than as note-takers. If the final product is not as pleasing to the eye as one could wish, this is not their fault; it indicates that much work remains to be done before this very promising topic reaches some degree of completion.
Princeton, January 15, 1961
A. WEll
TABLE OF CONTENTS
CHAPTER I. PRELIMINARIES ON ADELE-GEOMETRY 1.1.
Ade 1es
1.2.
Adele-spaces attached to algebraic varieties
1.3.
Restriction of the basic field
CHAPTER II.
4
TAMAGAWA MEASURES
2.1.
Prel iminaries
2.2.
The case of an algebraic variety
10 the local
measure
13
2.3.
The global measure and the convergence factors
21
2.4.
Algebraic groups and Tamagawa numbers
22
CHAPTER III.
THE LINEAR, PROJECTIVE AND SYMPLECTIC GROUPS
3.1.
The zeta-function of a central division algebra
30
3.2.
The projective group of a central division algebra
41
3.3.
Isogenies
43
3.4.
End of proof of Theorem 3.3.1.
central simple
algebras
47
3.5.
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