Arithmetic Groups

169 Pages·1980·3.005 MB·English

by  James E. Humphreys (auth.)

Lecture Notes ni
Mathematics
Edited by .A Dold dna .B nnamkcE
789
semaJ .E syerhpmuH
citemhtirA Groups
galreV-regnirpS
Berlin Heidelberg New kroY 1980
Author
James E. Humphreys
Department of Mathematics & Statistics
GRC Tower
University of Massachusetts
Amherst, MA 01003
USA
AMS Subject Classifications (1980): 10 D07, 20 G 25, 20 G 30, 20G 35,
20H05, 22E40
ISBN 3-540-09972-7 Springer-Verlag Berlin Heidelberg NewYork
ISBN 0-387-09972-7 Springer-Verlag NewYork Heidelberg Berlin
yrarbiL of in Publication Cataloging Congress .ataD ,syerhpmuH semaJ .E Arithmetic
in notes (Lecture groups. scitamehtam ; 789) Bibliography: .p index. Includes .1 Linear
groups. algebraic .2 groups. Lie .I Title. L1 in notes Lecture Series: scitamehtam ; (Berlin)
.82L.3AQ .987 .on 789. .]171AQ[ s015 [5t2'.2] 22921-08
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PREFACE
An arithmetic group is (approximately) a discrete subgroup of a
Lie group defined by arithmetic properties - for example, Z in ~,
GL(n,Z) in GL(n,~), SL(n,Z) in SL(n,~). Such groups arise in a wide
variety of contextss modular functions, Fourier analysis, integral
equivalence of quadratic forms, locally symmetric spaces, etc. In
these notes I have attempted to develop in an elementary way several
of the underlying themes, illustrated by specific groups such as
those just mentioned. While no special knowledge of Lie groups or
algebraic groups is needed to appreciate these particular examples,
I have emphasized methods which carry over to a more general setting.
None of the theorems presented here is new. But by adopting an
elementary approach I hope to make the literature (notably Borel [5]
and Matsumoto [1~)appear somewhat less formidable.
Chapters i - III formulate some familiar number theory in the
setting of locally compact abelian groups and discrete subgroups
(following Cassels [1], cf. Weil [2~ and Goldstein [I]). Here the
relevant groups are the additive group and the multiplicative group,
taken over local and global fields - or over the ring of adeles of a
global field. One basic theme is the construction of a good funda-
mental domain for a discrete group inside a locally compact group,
e.g., ~ in ~, or the ring of integers K O of a number field K inside
~n (n the degree of K over Q), where a fundamental domain corresponds
to a parallelotope determined by an integral basis of K over Q. In
the framework of adeles or ideles such fundamental domains have nice
arithmetic interpretations. Another basic theme is strong approxi-
mation. These introductory chapters are not intended to be a first
course in number theory, so the proofs of a few well known theorems
are just sketched.
Chapters IV and V deal with general linear and special linear
groups, emphasizing "reduction theory" in the spirit of Borel [5].
Here one encounters approximations to fundamental domains (called
"Siegel sets") for GL(n,Z) in GL(n,~) and deduces, for example, the
finite presentability of GL(n,Z) or SL(n,E). The BN-pair (Tits
system) and Iwasawa decomposition are used heavily here. There is
also a brief introduction to adelic and p-adic groups.
Finally, Chapter VI recounts (in the special case of SL(n,~))
the approach of Matsumoto [I~ to the Congruence Subgroup Problem,
IV
via central extensions and " Steinberg symbols". Here adeles and
strong approximation play a key role, along with the Bruhat decomposi-
tion already treated in IV. Matsumoto's group-theoretic arguments,
done in detail, lead ultimately to the deep arithmetic results of
Moore [i~, which can only be summarized here. (It is only fair to
point out that SL(n,~) can be handled in a more self-contained way,
cf. Bass, Lazard, Serre [i~, Mennicke [1], and unpublished lectures of
Steinberg. Special linear and symplectic groups over other rings of
integers can also be handled more directly, cf. Bass, Milnor, Serre
[1]. My objective has been to indicate the most general setting in
which the Congruence Subgroup Problem has so far been investigated~
in this generality it has not been completely solved.)
The various chapters can be read almost independently, if the
reader is willing to follow up a few references. I have tried to make
the notation locally (if not always globally) consistent. Standard
symbols such as ,2 Q, ~, C are used, along with ~0 (resp. ~ ~ )0 for
the set of positive (resp. nonnegative) reals. If K is a field, K
denotes its multiplicative group.
Chapters I - V are a revision of notes published some years ago
by the Cour~nt Institute. Chapter VI is based partly on a course I
gave at the University of Massachusetts~ class notes written up by
the students were of great help to me. I am grateful to the National
Science Foundation for research support, and to Peg Bombardier for her
help in typing the manuscript.
J.E. Humphreys
CONTENTS
I. LOCALLY COMPACT GROUPS AND FIELDS
i. Haar measure
1.1 Existence and uniqueness . . . . . . . . . . . 1
1.2 Module of an automorphism . . . . . . . . . . . 2
1.3 Homogeneous spaces . . . . . . . . . . . . . . 3
.2 Local and global fields
2.1 Classification theorem . . . . . . . . . . . . 4
2.2 Structure of local fields . . . . . . . . . . . 5
Appendix: Review of number fields and completions . 6
3. Adele ring of a global field
3.1 Restricted topological products ........ 9
3.2 Adeies . . . . . . . . . . . . . . . . . . . . 10
II, THE ADDITIVE GROUP
4. The quotient ~K/K
4.1 The space K~ . . . . . . . . . . . . . . . . . li
4.2 Fundamental domain for K in ~K ........ 13
4.3 Product formula . . . . . . . . . . . . . . . . 14
5. Volume of fundamental domain
5.1 Normalized Haar measure . . . . . . . . . . . . 15
5.2 Volume calculation . . . . . . . . . . . . . . 16
5.3 Application: Fields of discriminant ~ i .... 16
6. Strong approximation
6.1 Chinese Remainder Theorem . . . . . . . . . . . 18
6.2 An important lemma . . . . . . . . . . . . . . 19
6.3 Main theorem . . . . . . . . . . . . . . . . . 20
III. THE MULTIPLICATIVE GROUP
7. Ideles
7.1 Idele topology . . . . . . . . . . . . . . . . 21
7.2 Special ideles . . . . . . . . . . . . . . . . 22
.8 Compactness theorem
0 *
8.1 Compactness of JK/K . . . . . . . . . . . . . 24
8.2 Applications: Class number and units of K 25
8.3 Fundamental domain . . . . . . . . . . . . . . 28
IV. GL n AND SL n (OVER [)
9. Example: The modular group . . . . . . . . . . . . 29
10. Siegel sets in GL(n,~)
10.1 lwasawa decomposition . . . . . . . . . . . . 31
10.2 Siegel sets . . . . . . . . . . . . . . . . . . 33
10,3 Minimum principle . . . . . . . . . . . . . . 34
VI
11. Applications
11.1 Siegel sets in SL(n,R) . . . . . . . . . . . . 36
11.2 Reduction of positive definite quadratic forms 39
12. BN-pairs
12.1 Axioms and Bruhat decomposition ........ 41
12.2 Parabolic subgroups . . . . . . . . . . . . . . 43
12.3 Conjugates of B by W . . . . . . . . . . . . . 46
12.4 Complements for GL n . . . . . . . . . . . . . . 51
13. Siegel property (and applications)
13.1 Siegel sets revisited . . . . . . . . . . . . . 53
13.2 Fundamental sets and Siegel property ..... 56
13.3 Proof of Harish-Chandra's theorem ....... 58
13.4 Finite presentation of ~ . . . . . . . . . . . 61
13.5 Corners and arithmetic groups . . . . . . . . . 64
V. GL n AND SL n (p-ADIC AND ADELIC GROUPS)
14. Adelic groups
14.1 Adelization of a linear group . . . . . . . . . 65
14.2 Class number . . . . . . . . . . . . . . . . . 66
14.3 Strong approximation . . . . . . . . . . . . . 68
14.4 Reduction theory . . . . . . . . . . . . . . . 70
15. SL 2 (over p-adic fields)
15.1 Infinite dihedral group . . . . . . . . . . . 71
15.2 Lattices in 2 K . . . . . . . . . . . . . . . . . 72
15.3 BN-pair in G . . . . . . . . . . . . . . . . . 73
15.4 Building attached to BN-pair ......... 77
15.5 lhara's theorem; maximal compact subgroups . . 78
Appendix: Graphs and free groups . . . . . . . . . . 81
V!. THI~ CONGRUENCE SUBGROUP PROBLEM
16. Reformulation of the problem
16.1 Topological groups . . . . . . . . . . . . . . 85
16.2 Subgroup topologies on SL(n,Q) and SL(n,~) . . 86
16.3 Review of topology . . . . . . . . . . . . . . 88
16.4 Profinite groups . . . . . . . . . . . . . . . 90
16.5 Completions of topological groups ...... 92
16.6 The congruence kernel . . . . . . . . . . . . 97
17. The congruence kernel of SL(n,~)
17.1 Some consequences of the invariant factor
theorem .... 98
17.2 Congruence subgroups and q-elementary subgroups 100
17.3 A finiteness lemma . . . . . . . . . . . . . . 103
17.4 Proof of the theorem . . . . . . . . . . . . . 104
Vii
17.5 The congruence kernel . . . . . . . . . . . . . 106
17.6 Universal property of the extension ...... 108
18. The Steinberg group
18.1 Generators and relations . . . . . . . . . . . 109
18.2 The upper unitriangular group . . . . . . . . . Ii0
18.3 The monomial group . . . . . . . . . . . . . . iii
18.4 Steinberg symbols . . . . . . . . . . . . . . . 112
18.5 Determination of A . . . . . . . . . . . . . . 114
18.6 Determination of Ker @ . . . . . . . . . . . . 115
18.7 Universal property . . . . . . . . . . . . . . 116
18.8 Properties of Steinberg symbols ........ 119
19. Matsumoto's theorem
19.1 Central extensions and eocycles ........ 121
19.2 Statement of the theorem . . . . . . . . . . . 123
19.3 The diagonal group . . . . . . . . . . . . . . 124
19.4 An auxiliary construction . . . . . . . . . . . 124
19.5 The monomial group . . . . . . . . . . . . . . 129
19.6 Conclusion of the proof . . . . . . . . . . . . 132
19.7 The big cell . . . . . . . . . . . . . . . . . 136
19.8 The topological case . . . . . . . . . . . . . 137
20. Moore's theory
20.1 Topological Steinberg symbols . . . . . . . . . 140
20.2 Local and global theorems . . . . . . . . . . . 142
20.3 Central extensions of locally compact groups . 143
20.4 The fundamental group in the local case .... 144
20.5 Restricted products . . . . . . . . . . . . . . 145
20.6 Relative coverings . . . . . . . . . . . . . . 146
20.7 The congruence kernel revisited ........ 148
SUGGESTIONS FOR FURTHER READING . . . . . . . . . . . . . . . 149
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . 15]
INDEX . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
I. LOCALLY COMPACT GROUPS AND FIELDS
Here we shall review briefly the approach to local fields based
on the use of Haar measure, together with the construction of the
adele ring of a number field. For a full treatment of these matters,
the reader can consult Chapter I of Weil [2].
§i. Haar measure
For the standard results mentioned below, see Halmos [i, Chap-
ter XI] or Bourbaki [3, Chapter 7].
i.i Existence and uniqueness
Let G be a locally compact topological group (each element of
G has a compact neighborhood, or equivalently, the identity element
e does). G acts on itself by left and right translations
Xx : g ~--> xg
-I
Px : g ~--> gx
(here the inverse insures that we get a hom~omorphism x >---~ ~ of G
into the group of homeomorphisms of the space G).
Define a left (resp. right) Haar measure ~ on G to be a non-
zero Borel measure invariant under all left (resp. right) translations.
This means, by definition, that ~ is nonzero, all Borel sets are
measurable, ~(C) < ~ for C compact, and ~(XxM) = ~(M) (resp.
~(px M) = ~(M)) for x ~ G, M c G measurable.
Remarks and examples.
(i) If ~ is a left Haar measure on G, ~ is a right Haar measure,
where ~(X) = ~(X -I) for all measurable X -I. (Check!)
~>0
(2) If ~ is a left Haar measure on G, c ~ , then c~ is
again such (so Haar measure, if it exists, cannot be absolutely
unique).
(3) If G is abelian, left Haar measure = right Haar measure.
2
)4( G = ~ or C (additive group): Lebesgue measure is a Haar
measure.
)5( On finite products, the product measure is again a Haar measure
(for example, on ~n or Cn).
)6( On the multiplicative group ~>0 , dx/x is a Haar measure.
(Verify this by integrating functions of compact support.)
)7( G may have essentially distinct left and right Haar measures.
(See Halmos, p. 256, for the standard example.)
THEOREM. Let G be locally compact. Then G has a left (hence
also a right) Haar measure, and (up to a positive multiple) only one.
Haar proved the existence part for G having a countable basis
of open sets (1933). Later von Neumann proved uniqueness for compact
G. The general case was completed by Weil and yon Neumann.
Exercise. G locally compact, ~ (left) Haar measure.
)a( G is discrete iff ~({e}) > 0
)b( G is compact iff ~(G) < ~.
When G is compact, one frequently (but not always) normalizes ~ so
that ~(G) = i.
1.2 Module of an automorphism
The uniqueness part of Theorem i.I is more useful than may
appear at first. Let G be locally compact, with left Haar measure
. Aut G denotes the group of automorphisms of G (as topological
group).
If ¢ ~ Aut G, and X c G with ~(X) measurable, set
~(X) = ~(~(X)). Since ~ preserves Borel sets and compact sets,
it is very easy to see that ~ is again a left Haar measure on G.
By uniqueness, v = (mod G ¢)~ , where mOdG ~ ¢ ~>0 (and this num-
ber is independent of the original choice of ~, again by unique-
ness). Call mod G ¢ the (left) module of ~ .
-i
Example. Let ¢ = Int x : g ~ xgx (x e G). Here write
mod G ¢ = AG(X), so AG: G ÷ ~>0 is a function, called the module of
.G If G = A ,I call G unimodular (this means that left Haar
measure on G is also right Haar measure).
Exercises and examples.
(a) We could also have defined a right module of ¢.
Prove that this equals mod G ¢.
(b) mod G ¢ -mod = ~ G mOdG(¢O~)
)c( An abelian group is unimodular.
)d( Any automorphism of a discrete or compact group has module ,I
so such groups are unimodular.
(e) Any semisimple or nilpotent Lie group is unimodular.
Besides the example ¢ = Int x, another sort of automorphism
and its module will arise in §2 when we discuss locally compact
fields.
1.3 Homogeneous spaces
THEOREM. Let G be locally compact, H a closed subgroup of
G. Then there exists a G-invariant nonzero Borel measure on the
homogeneous space G/H iff the function AG, restricted to H,
equals AH; in this case, such a measure is unique up to a positive
multiple.
When G is abelian, or G is semisimple and H discrete, etc.,
the hypothesis will be fulfilled. It is cases like these that will
occupy us later.
§2. Local and global fields
Here and in subsequent sections we are following the approach of
Cassels [i] (cf. also Weil [2, Part I], Lang El, Chapter VIII, Gold-
stein [i, Part i]).