An introduction to arithmetic groups (via group schemes)

An introduction to arithmetic groups (via group schemes)

Ste↵en Kionke

25.06.2020

First definition of arithmetic groups Group schemes

Definition of arithmetic groups via group schemes

Examples of arithmetic groups

a

SL _n ( Z )

✓ SL

( R )

b

SL _n ( Z [ p 5])

✓ SL

( C )

c

H ₃ ( Z ) = { 0

@

1 x z 0 1 y 0 0 1

1

A | x, y, z 2 Z}

✓ H

( R )

d

U (p, q)( Z ) = { g 2 GL n ( Z [i]) | g ^T I p,q g = I p,q }

✓ U (p, q)

e

The unit group ⇤

✓ GL

( R )

where ⇤ is the ring

⇤ = Z i Z j Z ij Z with i ² = 2, j ² = 5, ij = ji.

a

SL _n ( Z ) ✓ SL

( R )

b

SL _n ( Z [ p

5]) ✓ SL

( C )

c

H ₃ ( Z ) = { 0

@

1 x z 0 1 y 0 0 1

1

A | x, y, z 2 Z} ✓ H

( R )

d

U (p, q)( Z ) = { g 2 GL n ( Z [i]) | g ^T I p,q g = I p,q } ✓ U (p, q)

e

The unit group ⇤

✓ GL

( R ) where ⇤ is the ring

⇤ = Z i Z j Z ij Z with i ² = 2, j ² = 5, ij = ji.

A first definition

Definition:

Let G ✓ GL n ( C ) be a Zariski closed subgroup defined over Q . An arithmetic subgroup of G is a subgroup

✓ G which is commensurable to G \ GL _n ( Z ).

commensurable:

vawüshing^Set

of polynom

with ^a

I coeff

n

Gln PQ ⁾

# =P

D= GAGEN AB ^E H comnenswabk

if An B has finite ^index

^{A. FB}

Group schemes

R: commutative unital ring

Alg _R : Category of commutative R-algebras

Definition: An affine group scheme (of finite type over R) is a covariant functor

G : Alg

R ! Grp

i.e., there is a natural equivalence G ! Hom _Alg

R

( O G , · ).

At

GIA )

Group schemes

R: commutative unital ring

Alg _R : Category of commutative R-algebras

Definition: An affine group scheme (of finite type over R) is a covariant functor

G : Alg

R ! Grp

which is representable by a finitely generated R-algebra O G , i.e., there is a natural equivalence G ! Hom _Alg

R

( O G , · ).

asfmetos sets

GAN ^E- Hohn

getan

^A)

f ^:*

^{B GA )} ! ^Ct _{IF I}

GCB) ¥ ^Hom

g( ^&

^{B) fod}

(1) The additive group G a (over R):

G a : A 7! (A, +)

Representable?

QGA

^{RET ]} Hom

g.

( ^RETJ

^{A) Es A}

×

LIT )

Examples

(2) The multiplicative group G m (over R):

G m : A 7! (A

, · ) Representable?

0am

^RET

^T

]

Hom ^IRETT

^]

^A)

^Ä

AGR L ^Ins LCT )

(3) The special linear group SL _n (over R):

SL _n : A 7! SL _n (A)

Representable?

Ogg

=

R [ Tijlicj ^EH MY _{( ldetkij ) )}

1)

Hohn , Sla

A)

SKA )

d ^t

( ^LC ₎ )

ij

Homomorphisms of group schemes

G, H affine group schemes over R.

Definition: A homomorphism ' : G ! H is a natural transformation of functors.

GLA ₎ ¥ _HCH )

f

^A-

_{ ^{Gift 0} _f. ^{HH )}

GCB ₎

HCB )

B

East ^:[ Yonedäslemna ^] Ü ^:O

OG

%

GCAIEHonl9.tt/-sHonlQtAEHH)LhXoCf

': G ^m ! SL ₂

' _A : A

! SL ₂ (A) with a 7!

✓ a 0 0 a ¹

◆

On coordinate rings?

R [ Trutz

Fritze ]

# Terme

)

→

RITT

]

Tu

^T

Tzz ^{1- T}

- t

Tietze ^{Im ⑨}

Coordinates

G an affine group scheme.

Definition:

A set of coordinates is an ordered tuple c = (t ₁ , . . . , t _n ) of elements of O G such that t 1 , . . . , t n generate O G .

R[T ₁ , . . . , T _n ]/I _c

⁼ ! O G

Coordinate map:

c,A : G(A)

⁼ ! Hom _Alg

R

( O G , A) ! V _A (I _c ) ✓ A ⁿ Ti ^ht

⇐

a.

(Ltte

Hlt

)

# ^{( Ic )}

^{{ las}

^{an IEÄ I fan}

.ae/--ofoaHfEIe _}

Comultiplication:

: O G ! O G ⌦ R O G

Coinversion:

I : O G ! O G

Counit:

": O ! R Satisfy axioms dual to the group axioms, e.g.,

G(A) G(A) ⇥ G(A) O G O G ⌦ R O G

{ 1 } G(A) R O G

mult

"

Link^id

)

left

inverse

The counit of a group scheme

The counit of G is the homomorphism ": O G ! R corresponding to the unit 1 2 G(R) via

G(R)

⁼ ! Hom _Alg

R

( O G , R).

Every R-algebra A is equipped with the structure morphism

◆ : R ! A

Usually ◆ " is also called counit and denoted by ".

1-

^

GCR ₎ Ü _{GIA ) #}

EI TE

Hom

⁾

^Haha

{

G a group scheme over R.

R ✓ S a ring extension.

Observation:

The functor

E _S/R (G) : Alg _S ! Grp

E _S/R (G)(A) = G(A | R ) is an affine group scheme over S.

K E G

cousider A

←

as R^-

algebra

CG )

=

5¥ ^!

Linear algebraic groups

K a field.

Definition:

A linear algebraic group over K is an affine group scheme over K such that O G has no nilpotent elements.

Remark: char(K) = 0 = ) the ring O G is reduced.

§

^veduced

"

Let G be a linear algebraic group over Q . Definition:

An integral form of G is a group scheme G ₀ over Z with an isomorphism

E

_/

(G ₀ ) ⇠ = G.

Definition:

A subgroup ✓ G( Q ) is arithmetic if it is commensurable to

G ₀ ( Z ) for some integral form G ₀ of G.

An example

Quaternion algebra:

D = (2, 5 | Q ) = Q Q i Q j Q ij

with i ² = 2, j ² = 5, ij = ji.

Linear algebraic group over Q :

G(A) = (A ⌦

D)

Integral form:

⇤ = Z i Z j Z ij Z G 0 (A) = (A ⌦

⇤)

Exercise

Check that

group

Scheine

Quaternion algebra:

D = (2, 5 | Q ) = Q Q i Q j Q ij

with i ² = 2, j ² = 5, ij = ji.

Linear algebraic group over Q :

G(A) = (A ⌦

D)

Integral form:

⇤ = Z i Z j Z ij Z G 0 (A) = (A ⌦

⇤)

Q ED

Golz )

Ä

^group of DX

Relation to first definition?

Fact:

Let G be a linear algebraic group over K. There is a “closed embedding” G , ! GL _n .

Proposition:

Let G be a linear algebraic group over Q and ' : G , ! GL n a closed embedding. Then there is an integral form G ₀ of G such that

' ¹ (GL n ( Z )) = G 0 ( Z ).

9

^G

^Gla

ewbeddiws if

OG

Gnade ⁾

Two results

Let G be a linear algebraic group over Q . Theorem 1:

If G ₀ , G ₁ are integral forms of G, then G ₀ ( Z ) and G ₁ ( Z ) are commensurable as subgroups of G( Q ).

Arithmetic groups are residually finite.

Two results

Let G be a linear algebraic group over Q . Theorem 1:

If G ₀ , G ₁ are integral forms of G, then G ₀ ( Z ) and G ₁ ( Z ) are commensurable as subgroups of G( Q ).

Lemma 2:

Arithmetic groups are residually finite.

"

Ij¥Äj ^{p :p}

^{F Hinte )}

ßC ^{g) EFAF}

Oberere

Sufticiat ^{to prove that GER )}

^{ideal }} finite

G a group scheme over Z , m 2 N

⇡ _m : Z ! Z /m Z

G(⇡ m ) : G( Z ) ! G( Z /m Z ) Observation: G( Z /m Z ) is finite.

Principal congruence subgroup:

G( Z , m) = ker(G(⇡ _m ))  f.i. G( Z ).

finite

1

% :* , GC %) ^Es 4,4 ⁾ EY

⁾

Proof of Lemma 2

Lemma 2: Arithmetic groups are residually finite.

JE ^{Gfk )} zt ¹

Cousin

g

^Oa

^Z

JFE

gtx ^{) # ECX)} _XEOG ^fasane

JG ^{) # Ecx ) modm} ( ^fern

¹⁾

Gltm ) G)

^Im 0J ^{# Tao {}

^{1 C-} Gtz )

Proof of Theorem 1

Theorem 1: If G ₀ , G ₁ are integral forms of G, then G ₀ ( Z ) and G ₁ ( Z ) are commensurable as subgroups of G( Q ).

Aim:

G ₀ ( Z ) \ G ₁ ( Z ) ◆ G ₀ ( Z , b) for some b 2 N

For simplicity we assume O G

, O G

✓ O G .

GdzlEG.ca

^EGCG

UI

Gfk ⁾

Simikrlg

ZGCZ.br

)

Proof of Theorem 1

Theorem 1: If G ₀ , G ₁ are integral forms of G, then G ₀ ( Z ) and G ₁ ( Z ) are commensurable as subgroups of G( Q ).

Aim:

G ₀ ( Z ) \ G ₁ ( Z ) ◆ G ₀ ( Z , b) for some b 2 N

We know Q ⌦

O G

⇠ = O G ⇠ = Q ⌦

O G

.

For simplicity we assume O G

, O G

✓ O G . They geniale

OG

Q

algebra Observation ^{1 EGOCZ )}

GfK ^{) EGCG )}

E

OG

Q

ECOG

) ^EZ

E. ( Oau ) ^EZ

Choose coordinates

f 1 , . . . , f _k 2 O G

with "(f i ) = 0 g 1 , . . . , g

2 O G

with "(g j ) = 0

Since f ₁ , . . . , f _k generate O G as Q -algebra, there are polynomials p 1 , . . . , p

2 Q [X 1 , . . . , X _k ] s.t.

p _j (f ₁ , . . . , f _k ) = g _j for all j 2 { 1, . . . , ` }

L ¥

Elf ^;)

Obst ^{pj her anstaut fern 0}

0

Ecgj ⁾

Elpjffe

fa ⁾ )

pjfdfel

^{Eda ) )}

=p ; 90

Proof of Theorem 1

b 2 N : a common denominator of all coefficients of p ₁ , . . . , p

. Claim:

G ₀ ( Z , b) ✓ G ₀ ( Z ) \ G ₁ ( Z )

f- ^{Gothia )} _g

⁶ ggf ^EZ

ycxIEECxlmodbbfarakxefotoshovi.gg ^{a) EZ ( GEEK )}

ie.

glgjk-kfa.at/jJGiI=Jlpilfa--fnI)=pjttfd....JfzIIEZ

_Eöödb

EI