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```Definition of arithmetic subgroup
Arithmetic subgroups of SL(n, R)
Interested in integer points of a group G вЉ† SL(N, R):
elements of G whose matrix entries are intвЂ™s.
GZ = G в€© SL(N, Z).
Dave Witte Morris
http://people.uleth.ca/в€јdave.morris
Dave.Morris@uleth.ca
Abstract
SL(2, Z) is an вЂњarithmeticвЂќ subgroup of SL(2, R). The other
arithmetic subgroups are not as obvious, but they can be
constructed by using quaternion algebras. Replacing the
quaternion algebras with larger division algebras yields many
arithmetic subgroups of SL(n, R), with n > 2. In fact, a
calculation of group cohomology shows that the only other
way to construct arithmetic subgroups of SL(n, R) is by using
unitary groups.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
1 / 38
Example
SO(1, n)Z is a arithmetic subgroup of SO(1, n).
where
I1,n =
пЈЇ
пЈЇ
пЈЇ
пЈЇ
пЈЇ
пЈЇ
пЈ°
в€’1
в€’1
..
.
в€’1
Arithmetic subgroups of SL(n, R)
4 / 38
2 / 38
Jeonju, August 2013
5 / 38
a,b
In general,
Remark
a,b
If HQ is a division algebra
пїЅв€љ пїЅ (в€Ђx в‰ 0, в€ѓy, xy = 1),
a,b
b в‰ пїЅ в€’ aпїЅ)
then SL(n, HZ ) not commensвЂ™ble to SL(2n, Z).
D any (finite-dimensional) division algebra over Q,
and D вЉ— R пїЅ MatdГ—d (R),
в‡’ SL(n, DZ ) is an arith subgroup of SL(dn, R).
Summary: Some arithmetic subgroups of SL(n, R)
can be constructed from division algebras.
All the others come from unitary groups.
7 / 38
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
3 / 38
Example
An arithmetic subgroup of SL(n, C).
We need to embed SL(n, C) in some SL(N, R):
C пїЅ R2 , so Cn пїЅ R2n , so SL(n, C) пїЅ SL(2n, R).
Then SL(n, C)Z = SL(n, C) в€© SL(2n, Z).
SL(n, C)Z = SL(n, C) в€© SL(2n,
пїЅ Z) пїЅn пїЅ
пїЅn
= { gпїЅв€€ SL(n,пїЅ C) | g Z[i] = Z[i] }
= SL n, Z[i] .
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
6 / 38
How to find the Q-forms of SL(n, R)
is arith subgrp of SL(2n, R).
a ;
Jeonju, August 2013
Since SL(2n, Z) = { g в€€ SL(2n, R) | gZ2n = Z2n },
SL(1, HZ ) is an arithmetic subgroup of SL(2, R)
a,b
SL(n, HZ )
Arithmetic subgroups of SL(n, R)
2
For П•(a, b) = a + bi, we have
пїЅ П•(ZпїЅn) = Z[i],
2n
so Z is identified with Z[i] вЉ‚ Cn .
how to find all of the others
Arithmetic subgroups of SL(n, R)
Dave Witte Morris (Univ. of Lethbridge)
пїЅ
These lectures:
Dave Witte Morris (Univ. of Lethbridge)
[Borel & Harish-Chandra, 1962]
This depends on the identification П• : R2 в†’ C:
The obvious Q-form of SL(n, R) is SL(n, Q),
corresponding to the arith subgrp SL(n, Z).
(b not norm in Q
Choose square-free a, b в€€ Z .
a,b
HZ = ZI +пїЅZi + Zj + Zk
пїЅ вЉ‚ Mat
пїЅ 2Г—2 (R),
пїЅ
в€љ
a
0
0 1
в€љ
where i =
, j=
, k = ij.
0 в€’ a
b 0
SL(2, R) acts on Mat2Г—2 (R) by multiplication gA.
a,b
For П• : R4 пїЅ Mat2Г—2 (R) with П•(Z4 ) = HZ ,
a,b
a,b
SL(2, R)Z = { g в€€ SL(2, R) | gHZ = HZ }
пїЅ a,b пїЅГ—
a,b
= HZ
пїЅ SL(1, HZ ).
+
Jeonju, August 2013
Jeonju, August 2013
(up to isomorphism).
The quaternions H = Hв€’1,в€’1 can be embedded in Mat2Г—2 (C):
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ
i 0
0 1
0 i
1 пїЅ I, i пїЅ
, jпїЅ
, k пїЅ ij =
.
0 в€’i
в€’1 0
i 0
Arithmetic subgroups of SL(n, R)
Arithmetic subgroups of SL(n, R)
Equivalent if G is connected and [G, G] = G:
GQ is dense in G, where GQ = G в€© SL(N, Q).
Definition. Subgroup GQ is called a вЂњQ-formвЂќ of G.
(up to commensurability)
Exercise. Another arithmetic subgroup of SL(2, R).
Dave Witte Morris (Univ. of Lethbridge)
Dave Witte Morris (Univ. of Lethbridge)
More precisely, there are polynomials
f1 (x1,1 , . . . , xN,N ), . . . , fm (x1,1 , . . . , xN,N )
with
пїЅ in Q,
пїЅ coeп¬ѓcients
пїЅ
пїЅ f (g , . . . , g
(gi,j )
пїЅ k 1,1
N,N ) = 0,
пїЅ
s.t. G пїЅ
.
в€€ SL(N, R) пїЅ
for all k
в‡’ GZ is a lattice in G
is the same as finding all the Q-forms of G
пЈє
пЈє
пЈє
пЈє.
пЈє
пЈє
пЈ»
Jeonju, August 2013
Remark. We usually ignore finite groups.
G пїЅ H means some finite-index subgroup of G
is equal to some finite-index subgroup of H.
In other words, G and H are commensurable.
Finding all the arithmetic subgroups of G
Write g I1,n g T = I1,n in terms of mat entries (gi,j ).
Obtain (n + 1)2 polynomial eqns, with coeп¬Ђs in Q.
Therefore SO(1, n) is defined over Q,
so SO(1, n)Z is an arithmetic subgroup.
Dave Witte Morris (Univ. of Lethbridge)
Example. SL(n, Z) is an arith subgroup of SL(n, R).
A Lie group G usually has many arithmetic subgrps,
because there are many embeddings G пїЅ SL(N, R),
which yield very diп¬Ђerent arithmetic subgroups.
Proof.
SO(1, n) = { g в€€ SL(n
+ 1, R) | g I1,nпЈ№g T = I1,n },
пЈ®
1
Definition
Spse G вЉ† SL(N, R) (and a technical condвЂ™n is satisfied).
Then GZ is an arithmetic subgroup of G.
The technical condition
Need to assume G is defined over Q:
G is defвЂ™d by polynomial eqвЂ™ns with ratвЂ™l coeп¬Ђs.
8 / 38
Let G = SL(n, R). Suppose ПЃ : G пїЅ SL(N, R),
such that ПЃ(G) is defined over Q.
Find ПЃ(G)
Galois theory:
пїЅ
пїЅ Q by using
пїЅ
Q = z в€€ C пїЅ Пѓ (z) = z, в€ЂПѓ в€€ Gal(C/Q) .
пїЅ
пїЅ
пїЅ
пїЅ
Пѓ (h) = h,
пїЅ
SL(N, Q) = h в€€ SL(N, C)пїЅ
.
пїЅ в€ЂПѓ в€€ Gal(C/Q)
ПЃ(G)QпїЅ= ПЃ(G)
= пїЅ ПЃ(g) пїЅ ПѓпїЅ ПЃ(g) = ПЃ(g), в€ЂПѓ в€€ Gal(C/Q)
пїЅ (g) = g, в€ЂПѓ в€€ Gal(C/Q)
пїЅ g в€€ GC пїЅ Пѓ
пїЅ = ПЃ в€’1 Пѓ ПЃ : GC в†’ GC .
where Пѓ
Every Q-form of G is the fixed points of
an action of Gal(C/Q) on GC .
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
9 / 38
Lecture 2
Every Q-form of G is the fixed points of
an action of Gal(C/Q) on GC .
Recall
What are the arithmetic subgrps of G = SL(n, R)?
Embed G пїЅ SL(N, R).
Same problem:
Find GQ .
Find GZ = G в€© SL(N, Z).
(вЂњQ-formвЂќ)
Every Q-form of G is the fixed points of
an action of Gal(C/Q) on GC .
пїЅ
пїЅ
SL(n, Q) = g в€€ GC пїЅ в€ЂПѓ в€€ Gal(C/Q), Пѓ (g) = g .
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
10 / 38
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
Q-form GQ
2
13 / 38
Kernel of О± is a subgroup of index 2 in Gal(C/Q),
пїЅв€љ пїЅ
so fixed field is a quadratic extension Q r of Q.
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
Jeonju, August 2013
11 / 38
пїЅ
пїЅ
пїЅ
О± в€€ H 1 Gal(C/Q), Aut(GC )
16 / 38
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
14 / 38
Jeonju, August 2013
12 / 38
Fact. Every C-linear aut of algebra MatnГ—n (C) is inner
вЂ” it is conjugation by
пїЅ a matrix in
пїЅ GL(n, C).
Scalars act trivially: Aut MatnГ—n (C) = PSL(n, C).
пїЅ
пїЅ
H 1 Gal(C/Q), PSL(n, C)
пїЅ
пїЅ
пїЅпїЅ
= H 1 Gal(C/Q), Aut MatnГ—n (C)
= { Q-forms of MatnГ—n (C) }
пїЅ
пїЅ
algebras A over Q,
=
.
such that A вЉ— C пїЅ MatnГ—n (C)
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
15 / 38
Corrections
Intermediate case
We seem to have shown that all arithmetic subgroups of
SL(n, R) can be constructed from either division algebras
(Case 1) or unitary groups (Case 2). However, the discussion
in Case 2 assumes that the restriction of О± to the kernel of О±
is trivial. If we remove this restriction,
пїЅ
пїЅв€љthen,
пїЅпїЅ by the argument
of Case 1, the cocycle from Gal C/Q r
into PSL(n,
пїЅв€љ пїЅC)
yields a simple algebra Matk (D)
center is Q r . The
пїЅв€љ whose
пїЅ
Galois automorphism О· of Q r can be extended to an
пїЅ of D. Then, for some A в€€ Matk (D), the
anti-automorphism О·
corresponding
пїЅ
пїЅ Q-form
пїЅ isпїЅ
пїЅ
пїЅ
пїЅ D = g в€€ SL(k, D) пїЅ gA(g О·пїЅ )T = A .
GQ = SU A, О·;
Note that this Q-form is obtained by combining unitary
groups with division algebras.
and A = I, this means gg в€— = I,
so g в€€ SU(n)Q[в€љr ] .
(unitary group)
If A = I m,n = diag(1, 1, . . . , 1, в€’ 1, в€’1, . . . , в€’1),
пїЅAпїЅ
so z
zв€— = |z1 |2 + В· В· В· + |zm |2
в€’ |zm+1 |2 в€’ В· В· В· в€’ |zm+n |2 ,
then g в€€ SU(m, n)Q[в€љr ] .
пїЅ
пїЅв€љ пїЅпїЅ
In general, we have g в€€ SU A, О·; Q r .
Arithmetic subgroups of SL(n, R)
Arithmetic subgroups of SL(n, R)
(and the center of D must be Q)
пїЅ пїЅв€љ пїЅ пїЅ
Let Gal Q r /Q = {1, О·}, so О· в€‰ ker О±.
Then О±О· = (conj by A) П‰ for some A в€€ GL(n, R).
пїЅ
пїЅв€’1 в€’1
пїЅ
We have g = О·(g)
= О±О· О·(g) = A (О·g)T
A ,
so g A (О·g)T = A.
Dave Witte Morris (Univ. of Lethbridge)
Dave Witte Morris (Univ. of Lethbridge)
The corresponding Q-form GQ is SL(k, D).
пїЅ
пїЅв€љ пїЅпїЅ
GQ вЉ† SL n, Q r
If О· =
If X is an algebraic object defined over Q, then
пїЅ
пїЅ
H 1 Gal(C/Q), Aut(XC )
1в€’1 пїЅ
в†ђв†’ Q-forms of X }
пїЅ
пїЅ
Q-isomorphism classes of
=
.
Q-defined objects whose
C-points are isomorphic to XC
A must be simple, so, by WedderburnвЂ™s Theorem,
A пїЅ Matk (D), where D is a division alg over Q.
Image of О± пїЅвЉ† PSL(n, C).
Dave Witte Morris (Univ. of Lethbridge)
H 1 Gal(C/Q), Aut(GC ) .
пїЅ
пїЅ
Case 1. Assume О± в€€ H 1 Gal(C/Q), PSL(n, C) .
We consider two cases:
пїЅ
пїЅ
1
О± в€€ H 1 Gal(C/Q), PSL(n, C) .
Case 2. Assume image of О± пїЅвЉ† PSL(n, C).
О± induces nontrivial
О±О± : Gal(C/Q) в†’ Aut Out(GC ) пїЅ Z2 .
Action of Gal(C/Q) on Z2 is trivial, so О± is a homo.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Fact. Only outer automorphism of GC = SL(n, C) is
transpose-inverse (П‰(g) = (g T )в€’1 ).
So Aut(GC ) = PSL(n, C) пїЅ пїЅП‰пїЅ.
пїЅ
пїЅ
Case 1. Assume О± в€€ H 1 Gal(C/Q), PSL(n, C) .
Consider any g в€€ GQ = { g в€€ GC | g ПѓпїЅ = g, в€ЂПѓ }.
For simplicity, assume О± is trivial on ker О±.
(If not, there is a division algebra involved.)
пїЅ
пїЅв€љ пїЅпїЅ
If Пѓ в€€ Gal C/Q r
= ker О±, then О±Пѓ is trivial.
пїЅ , so g Пѓ = g ПѓпїЅ = g.
This means
пїЅ
пїЅв€љПѓ =
пїЅпїЅПѓ
So g в€€ SL n, Q r .
Note. О±Пѓ = ПЃ в€’1 Пѓ ПЃ Пѓ в€’1 = ПЃ в€’1 ПѓПЃ looks like a cobdry
so it is a 1-cocycle.
пїЅ
пїЅ
So it defines an element of H 1 Gal(C/Q), Aut(GC ) .
We will find all the
R),
пїЅ Q-forms of SL(n,
пїЅ
пїЅпїЅ
by calculating H 1 Gal(C/Q), Aut SL(n, C) .
пїЅ
пїЅ
H 1 Gal(C/Q), SL(n, C) = 0
пїЅ
пїЅ
Warning. H 1 Gal(C/Q), PSL(n, C) в‰ 0.
1-cocycle пїЅ Оґ1 c = 0 пїЅ c(gh) = c(g) + g c(h)
Dave Witte Morris (Univ. of Lethbridge)
Finding the arithmetic subgrps of G amounts to
calculating
пїЅ the вЂњGalois cohomology
пїЅ setвЂќ
This is a special case of a fairly general principle:
Q-forms from Galois cohomology
Example
Suppose V1 and V2 are two vector spaces over Q,
and they are isomorphic over C.
(I.e., V1 вЉ— C пїЅ V2 вЉ— C.)
Then dim V1 = dim V2 ,
so V1 and V2 are isomorphic over Q.
Thus, the пїЅQ-form of any vector
пїЅ space is unique,
so H 1 Gal(C/Q), Aut(VC ) = 0,
for any vector
пїЅ space V over Q. пїЅ
In other words, H 1 Gal(C/Q), GL(n, C) = 0.
Similarly:
Group cohomology
Function c : О“ k в†’ A is k-cochain в€€ C k (О“ ; A).
Coboundary Оґk : C k (О“ ; A) в†’ C k+1 (О“ ; A)
Оґ0 a(g) = ga в€’ a
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ (g) = g .
GQ = g в€€ GC пїЅ в€ЂПѓ в€€ Gal(C/Q), Пѓ
пїЅ Пѓ в€’1 : GC в†’ GC . (continuous automorphism)
Let О±Пѓ = Пѓ
So О±Пѓ в€€ Aut(GC ). Thus, О± : Gal(C/Q) в†’ Aut(GC ).
GQ пїЅ [О±
between
пїЅ Пѓ ] provides a 1-1 correspondence
пїЅ
H 1 Gal(C/Q), Aut(GC ) and the set of Q-forms.
Jeonju, August 2013
17 / 38
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
18 / 38
SL(n, R) vs. SL(n, C)
Cocompact arithmetic subgroups
One more technique (familiar from the case of SL(2, R)) is
needed in order to obtain all of the arithmetic subgroups of
SL(n, R), because the above techniques do not suп¬ѓce to
construct some cocompact examples. The key point is that
we need to slightly extend the definition of an arithmetic
subgroup. Namely, instead of requiring О“ to be the integer
points of G itself, it may be necessary to choose a compact
group K, such that G Г— K is defined over Q, and allow О“ to be
the projection of (G Г— K)Z to G. Because of this, GQ can be a
unitary group over a (totally real) extension of Q, rather than
over Q itself. In other words, we need to consider arithmetic
subgroups obtained from вЂњRestriction of ScalarsвЂќ.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
19 / 38
Why unitary groups are arithmetic
Proposition
пїЅв€љ пїЅ
О· = Galois automorphism of Q r в‰ Q,
+
where r в€€ ZпїЅ is square-free,
в€љ пїЅ
в€љ
which means О· a + b r = a в€’ b r ,
пЈ®
пЈ№
пЈ®
пЈ№
1
1
пЈЇ
пЈє
пЈЇ
пЈє
J = пЈ° 1 пЈ» or, if you prefer, J = пЈ° в€’1
пЈ»
в€’1
1
пїЅ
пїЅв€љ пїЅпїЅ
О“r = SU J, О·; QZ r
пїЅ
пїЅ
пїЅ
пїЅв€љ пїЅпїЅ пїЅ
пїЅ
пїЅ gJ(О·g)T = J .
= g в€€ SL 3, Z r
Then О“r is an arithmetic subgroup of SL(3, R).
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
22 / 38
пїЅ = { (g, h) в€€ G Г— G | gJhT = J }
G
пїЅ пїЅ G.
Each g determines a unique h, so G
пїЅ
We may let ПЃ(G) = G.
в€љ
в€љ
(1, 1) and ( r , в€’ r ) are linearly independent,
so they form a basis of R2 .
6
Thus, в€ѓ invertible
linear trans пїЅof R6 that maps
пїЅ
пїЅв€љ пїЅZпїЅ to
О·
О·
О· пїЅ
О› = (x, y, z, x , y , z ) x, y, z в€€ Z r .
Since SL(6, Z) = { a в€€ SL(6, R) | aZ6 = Z6 },
this means we may pretend (after a change of basis)
(G Г— G) в€© SL(6, Z)
=пїЅ
{ (g, h) в€€пїЅ G Г— G |пїЅ(g, h)О› =пїЅпїЅ
О›}
пїЅв€љ пїЅ
пїЅ
= (g, g О· ) пїЅ g в€€ SL 3, Z r
.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
25 / 38
To discuss Galois cohomology, we replaced R with the
algebraically closed field C. Thus, some of the groups we
found might not be Q-forms of SL(n, R) (although we know
that theirпїЅ complexification
is SL(3, C)). For example, if
пїЅв€љ пїЅпїЅ
GQ = SU J, О·; Q r , and r < 0, then GR is SU(2, 1), not
SL(3, R).
In practice, one can determine which of the groups we
constructed are Q-forms of SL(3, R).
Abstractly, SL(3, R) is an R-form of SL(3, C), so, by the
general principle,
it is represented
пїЅ
пїЅ by a cohomology
class ОІ в€€ H 1 Gal(C/R), Aut(GC ) . There is a natural
restriction
пїЅ homomorphism пїЅ
пїЅ
пїЅ
r : H 1 Gal(C/Q), Aut(GC ) в†’ H 1 Gal(C/R), Aut(GC ) ,
and the Q-forms of SL(3, R) are represented by the
elements of r в€’1 (ОІ).
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
20 / 38
Theorem (Weil 1960 (or Siegel earlier?))
Suppose О“ is an arithmetic subgroup of G = SL(3, R).
If G/О“ is not compact, then О“ is commensurable to
either SL(3, Z) or О“r , for some r .
C vs. Q
For Galois cohomology, we should really be using
theпїЅalgebraic closure
пїЅ Q of Q, instead of C, and
H 1 Gal(Q/Q), GQпїЅ is defined toпїЅbe the natural limit
of the groups H 1 Gal(F /Q), GQ , where F ranges
over all finite extensions of Q.
Dave Witte Morris (Univ. of Lethbridge)
(But diп¬Ђerent AвЂ™s sometimes give the same Q-form.)
Let
Jeonju, August 2013
23 / 38
Want: ПЃ(G) в€© SL(6, Z) = ПЃ(О“r ).
пїЅ = { (g, h) в€€ G Г— G | gJhT = J },
Know: ПЃ(G) = G
and (G Г— G)
пїЅ
пїЅв€љ пїЅпїЅ пїЅ
пїЅ
= (g, g О· ) пїЅ g в€€ SL 3, Z r
.
Then
ПЃ(О“r ) = { (g, g О· ) | g в€€ О“r },
and
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ
пїЅ g в€€ SL 3, Z в€љr ,
пїЅ
= (g, g О· )пїЅ
пїЅ
gJ(g О· )T = J
пїЅ
пїЅ
пїЅ
пїЅв€љ пїЅпїЅпїЅ
О· пїЅ
= (g, g ) пїЅ g в€€ SU J, О·; Z r
= ПЃ(О“r ).
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
21 / 38
Construct embedding ПЃ : G пїЅ SL(6, R),
such that
ПЃ(G) в€© SL(6, Z) = ПЃ(О“r ).
Note:
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
Want: О“r is an arithmetic subgroup of G = SL(3, R)
Remark
The value of r is unique: if r1 в‰ r2 , then
no finite-index subgroup of О“r1 is isomorphic to
a finite-index subgroup of О“r2 .
пїЅв€љ пїЅ
пїЅв€љ пїЅ
Q r1 в‰ Q r2 пїЅв‡’ ker О±1 в‰ ker О±2
пїЅв‡’ О±1 в‰ О±2 пїЅв‡’ О±1 пїЅв€ј О±2
пїЅв‡’ Q-forms are diп¬Ђerent.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
пЈ®
пЈ№
в€— в€— в€—
пЈЇ
пЈє
пЈЇв€— в€— в€—
пЈє
пЈЇ
пЈє
пЈЇв€— в€— в€—
пЈє
пЈє вЉ‚ SL(6, R).
GГ—G =пЈЇ
пЈЇ
пЈє
в€—
в€—
в€—
пЈЇ
пЈє
пЈЇ
в€— в€— в€—пЈє
пЈ°
пЈ»
в€— в€— в€—
пїЅ = { (g, h) в€€ G Г— G | gJhT = J }.
G
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
24 / 38
ПЃ(G) в€© SL(6, Z) = ПЃ(О“r )
Conclude О“r is arith subgrp if ПЃ(G) defвЂ™d over Q.
Can be done directly (find polynomials with coeп¬Ђs in Q),
but it is confusing вЂ“ need to work in a strange basis.
26 / 38
Instead, we will verify that GQ is dense in G.
пїЅ
пїЅв€љ пїЅпїЅ
GZ = О“r = SU J, О·; Z r
пїЅ
пїЅ
пїЅ
пїЅв€љ пїЅпїЅ пїЅ
пїЅ
пїЅ gJ(g О· )T = J .
= g в€€ SL 3, Z r
пїЅ
пїЅв€љ пїЅпїЅ
в‡’ GQ = SU J, О·; Q r
пїЅ
пїЅ
пїЅ
пїЅв€љ пїЅпїЅ пїЅ
пїЅ
пїЅ gJ(g О· )T = J .
= g в€€ SL 3, Q r
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
27 / 38
пїЅ
пїЅ
пїЅв€љ пїЅпїЅ пїЅ
пїЅ
пїЅ gJ(g О· )T = J
g в€€ SL 3, Q r
пїЅ
пЈј
пїЅ
в€љ
xx
пЈґ
пїЅ
пЈґ
пїЅ
пїЅ
в€љ
x
t rв€’
пїЅ x в€€ Q r ,пЈґ
пЈґ
пЈґ
пїЅ
пЈЅ
2 пЈє
пЈєпїЅ
пЈєпїЅ
О·
вЉ‚ GQ .
1
в€’x
пЈєпїЅ
пЈґ
пЈґ
пЈ»пїЅ
пЈґ
tв€€Q
пЈґ
пЈґ
пїЅ
пЈѕ
пїЅ
1
пЈ®
пЈ№
1 в€— в€—
пЈЇ
пЈє
This is dense in U = пЈ° 1 в€—пЈ».
1
GQ also contains a dense subgroup of U T .
пЈ±пЈ®
пЈґ
пЈґ
пЈґ
1
пЈґ
пЈґ
пЈІпЈЇ
пЈЇ
пЈЇ
So пЈЇ
пЈґ
пЈґ
пЈ°
пЈґ
пЈґ
пЈґ
пЈі
О·пЈ№
(In fact, it is easy to verify that (GQ )T = GQ .)
Since пїЅU , U T пїЅ = SL(3, R) = G,
this implies GQ is dense in G.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
28 / 38
1. Division algebras
2. Unitary groups
пїЅв€љ пїЅ
F = Q r , and пїЅО·пїЅ = Gal(F
пїЅ /Q). пїЅ
A в€€ GL(n, F ) Hermitian (AО· )T = A
Can assume A = diag(a1 , . . . , an ), ai в€€ QГ— .
пїЅ
пїЅ
пїЅ пїЅ
пїЅ
GQ = SU A, О·; F = g в€€ SL(n, F ) пїЅ g A (g О· )T = A .
пїЅ
пїЅв€љ пїЅпїЅ
GZ пїЅ SU A, О·; Z r
Note: F is uniquely determined by GQ .
пїЅ
пїЅ
пїЅ
пїЅ
But SU A, О·; F пїЅ SU B, О·; F if B = XA(X О· )T .
в‡’ can make A diagonal
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
31 / 38
Type A: GC = SL(n, C).
Obvious R-form is SL(n,пїЅR).
пїЅ
Previous discussion: О± в€€ H 1 Gal(C/R), Aut(GC ) .
пїЅ
пїЅ
Case 1. О± в€€ H 1 Gal(C/R), Inn(GC ) . вЂњInner formвЂќ
Comes from division algebra over R.
Only division algebra is H: GR = SL(k, H).
вЂњOuter
Case 2. О± : Gal(C/R) в†’ Out(GC ) nontriv.
formвЂќ
Unitary group: GR = SU(m, n).
Combine unitary
пїЅв€љ пїЅ group with division algebra
over F = R r = C. пїЅ div alg over C.
Type B: GC = SO(n, C), with n odd.
Out(GC ) trivial, so all R-forms are inner.
GR = SO(p, q).
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
How to make arithmetic subgroups of SL(n, R)
1) division algebras Q-forms 2) unitary groups
3) combination of the two
1. Division algebras
division algebra D over Q, such that D вЉ— R пїЅ MatdГ—d (R)
пїЅ Q-form SL(n, D) of SL(dn, R).
Example: Quaternions
a,b
HQ = Q + Qi + Qj + Qk (i2 = a, j 2 = b, k =пїЅ ij =пїЅ в€’ji)
в€љ
= F + Fj
F = Q + Qi = Q a
j 2 = b, jx = x О· j, пїЅО·пїЅ = Gal(F /Q)
Division algebra if b is not a norm in F
b в‰ x x О· = пїЅ в€’ aпїЅ
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
29 / 38
пїЅв€љ пїЅ
F = Q r , and пїЅО·пїЅ = Gal(F
пїЅ /Q).
пїЅ
A в€€ GL(n, F ) Hermitian (AО· )T = A
пїЅ
пїЅ
пїЅ пїЅ
пїЅ
GQ = SU A, О·; F = g в€€ SL(n, F ) пїЅ g A (g О· )T = A .
3. Combine division algebras with unitary groups
пїЅв€љ пїЅ
F = Q r , and пїЅО·пїЅ = Gal(F /Q).
Division algebra D over F , such that
О·
О· О·
О· extends to antiaut О·пїЅ of D: (ab)
пїЅ =b a .
A в€€ GL(n, D) Hermitian (AО· )T = A
(Can assume A = diag(a1 , . . . , an ), ai в€€ D Г— , aiО· = ai .)
пїЅ
пїЅ
пїЅ
пїЅ
GQ = SU A, О·; D ,
GZ = SU A, О·; D Z
3. Combination
Dave Witte Morris (Univ. of Lethbridge)
a,b
Lecture 3
34 / 38
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
32 / 38
Type C:
GC = Sp(2n, C) = { g в€€ SL(2n,пЈ®C) | gJg T = J } пЈ№
where J =
Obvious R-form is Sp(2n, R).
пЈЇ
пЈЇ
пЈЇ
пЈЇ
пЈЇ
пЈЇ
пЈ°
1
.
пїЅ
..
GQ =
в€’1
в€’1
Out(GC ) is trivial (so forms inner), but can use H:
GR = SU(Ip,q , О·; H) = Sp(p, q).
О·(a + bi + cj + dk) = a в€’ bi в€’ cj в€’ dk
1
пЈє
пЈє
пЈє
пЈє.
пЈє
пЈє
пЈ»
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
How to make a division algebra over Q
D = F + F j + F j 2 + В· В· В· + F j dв€’1
(cyclic)
|F : Q| = d, j d = b в€€ Q, jx = x О· j, пїЅО·пїЅ = Gal(F /Q)
Eg. Choose p в‰Ў 1 (mod d). в€ѓ e в€€ ZГ—
в€’ 1)/d.
p of order (p
в€љ
пїЅ
2
(pв€’1)/d пїЅ
p
Let О¶ = 1, and F = Q О¶ e + О¶ e + В· В· В· + О¶ e
.
Division algebra if bm not a norm in F , for m < d.
2
dв€’1
bm в‰ x x О· x О· В· В· В· x О·
(m = 1 if d is prime)
GQ = SL(k, D).
35 / 38
GZ пїЅ SL(k, D Z ).
F = Q[П•], where П• is algвЂ™ic int, and let O = Z[П•].
Then DZ = O + Oj + Oj 2 + В· В· В· + Oj dв€’1 , if b в€€ Z.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
30 / 38
Same methods apply to other groups
Galois cohomology finds arithmetic subgroups of
any simple Lie group, not just SL(n, R).
Illustration: find the simple Lie groups (over R).
Simple Lie groups over C
Type A, B, C, D, E, F, G.
E, F, and G are exceptional groups вЂ” ignore.
A вЂ“ D are classical.
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
33 / 38
Arithmetic groups of classical type
GQ = SL(n, F ), SL(n, D), SU(A, О·; F ), SU(A, О· ; D)
в‡’ GR = SL(n, R), SL(n, C), SL(n, H), SU(p, q),
or product of these
or
GQ = SO(A, F )
Type D: GC = SO(2n, C). Obvious GR = SO(p, q).
Out(GC ) пїЅ Z2 (if 2n в‰ 8), but inner in O(2n, C).
Can use quaternions:
пїЅ H) = SO(n, H) = SOв€— (n).
GR = SU(I, О·;
пїЅ + bi + cj + dk) = a + bi в€’ cj + dk
О·(a
Dave Witte Morris (Univ. of Lethbridge)
HQ = F + F j,
|F : Q| = 2, j 2 = b в€€ Q, jx = x О· j, пїЅО·пїЅ = Gal(F /Q)
в‡’ GR = SO(p, q), SO(n, C) product
a,b
пїЅ HF )
GQ = SU(A, О·,
в‡’ GR = SO(n, H),
or preceding orthogonal groups or product
a,b
GQ = Sp(n, F ), SU(A, О·, HF )
в‡’ GR = Sp(n, R), Sp(n, C), Sp(p, q)
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
or product
Jeonju, August 2013
36 / 38
References
GQ = SL(n, F ), SL(n, D), SU(A, О·; F ), SO(A, F ),
пїЅ
пїЅ
a,b пїЅ
a,b пїЅ
пїЅ HF , Sp(n, F ), SU A, О·, HF
SU A, О·,
V. Platonov and A. Rapinchuk,
Algebraic Groups and Number Theory,
Academic Press, New York, 1994. MR1278263
This lists all Q-forms of classical simple Lie groups
(SL, SO, SU, Sp)
except some outer forms of
SO(8, C), SO(p, 8 в€’ p), SO(4, H).
пїЅ
пїЅпїЅ
пїЅ
Missing: пїЅOut(SO(8,
C) пїЅ = 6:
Image of О± can be Z3 or S3 .
вЂњtrialityвЂќ groups
(not trivial or Z2 )
(See В§2.2вЂ“В§2.3 for the use of Galois cohomology to
find Q-forms of simple algebraic groups.)
Dave Witte Morris,
Introduction to Arithmetic Groups (in preparation).
http://arxiv.org/abs/math/0106063
(Examples of Lattices chapter includes sections on lattices in
SL(3, R) and, more generally, SL(n, R).
Arithmetic Lattices in Classical Groups chapter describes the
arithmetic subgrps of all classical Lie grps, not just SL(n, R).
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
37 / 38
Dave Witte Morris (Univ. of Lethbridge)
Arithmetic subgroups of SL(n, R)
Jeonju, August 2013
38 / 38
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