Arithmetic Groups
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789 James E. Humphreys
Arithmetic Groups
Springer-Verlag Berlin Heidelberg New York 1980
Author James E. Humphreys Department of Mathematics & Statistics G R C Tower University of Massachusetts Amherst, M A 01003 USA
A M S Subject Classifications (1980): 10 D07, 20 G 25, 20 G 30, 2 0 G 35, 20H05, 22E40 ISBN 3-540-09972-7 Springer-Verlag Berlin Heidelberg NewYork ISBN 0-387-09972-7 Springer-Verlag NewYork Heidelberg Berlin Library of Congress Cataloging in Publication Data. Humphreys, James E. Arithmetic groups. (Lecture notes in mathematics ; 789) Bibliography: p. Includes index. 1. Linear algebraic groups. 2. Lie groups. I. Title. 1LSeries: Lecture notes in mathematics (Berlin) ; 789. QA3.L28. no. 789. [QA171].510s [5t2'.2] 80-12922 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 1980 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE An arithmetic
group is (approximately)
Lie group defined by arithmetic GL(n,Z)
in GL(n,~),
SL(n,Z)
variety of contextss equivalence
in SL(n,~).
modular
of quadratic
functions,
forms,
these notes I have attempted of the underlying those
themes,
just mentioned.
algebraic
groups
properties
a discrete
analysis,
locally symmetric
illustrated
spaces,
integral
etc.
by specific
groups such as
While no special knowledge
of Lie groups
these particular
here is new.
But by adopting
approach I hope to make the literature
and Matsumoto
[1~)appear
somewhat less formidable.
Chapters i - III formulate
(notably Borel
and discrete
[1], cf. Weil [2~ and Goldstein
taken over local and global fields
Here the
e.g.,
arithmetic
mation. course
determined
interpretations.
These introductory
in number theory,
Chapters
Here one encounters "Siegel sets")
"reduction theory"
approximations
finite presentability system)
are not intended
so the proofs
for GL(n,Z)
of GL(n,Z)
linear and special linear
in the spirit of Borel [5].
and deduces,
or SL(n,E).
domains
The BN-pair
are used heavily here.
to adelic and p-adic groups.
the approach
[I~ to the Congruence
Chapter VI recounts
(in the special
(called
for example,
also a brief introduction
of Matsumoto
to be a first
of a few well known theorems
to fundamental
in GL(n,~)
and Iwasawa decomposition
Finally,
In
domains have nice
Another basic theme is strong approxi-
chapters
IV and V deal with general
emphasizing
domain corresponds
by an integral basis of K over Q.
of adeles or ideles such fundamental
are just sketched. groups,
O K of a number field K inside
of K
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