adeleho2/doc/adele.pdf · Title adele.dvi Created Date 9/3/2009 6:02:24 AM

ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM

Hee Oh

Abstract. In this paper, we generalize Margulis’s S-arithmeticity theorem to the case

when S can be taken as an infinite set of primes. Let R be the set of all primes includinginfinite one ∞ and set Q∞ = R. Let S be any subset of R. For each p ∈ S, let Gp be

a connected semisimple adjoint Qp-group without any Qp-anisotropic factors and Dp ⊂

Gp(Qp) be a compact open subgroup for almost all finite prime p ∈ S. Let (GS , Dp)denote the restricted topological product of Gp(Qp)’s, p ∈ S with respect to Dp’s. Note

that if S is finite, (GS , Dp) =Q

p∈S Gp(Qp). We show that ifP

p∈S rank Qp(Gp) ≥ 2,

any irreducible lattice in (GS , Dp) is a rational lattice. We also present a criterion on the

collections Gp and Dp for (GS , Dp) to admit an irreducible lattice. In addition, we describediscrete subgroups of (GA, Dp) generated by lattices in a pair of opposite horospherical

subgroups.

1. Introduction

Let R denote the set of all prime numbers including the infinite prime ∞ and Rf the

set of finite prime numbers, i.e., Rf = R−∞. We set Q∞ = R. For each p ∈ R, let Gp

be a non-trivial connected semisimple algebraic Qp-group and for each p ∈ Rf , let Dp

be a compact open subgroup of Gp(Qp). The adele group of Gp, p ∈ R with respect to

Dp, p ∈ Rf is defined to be the restricted topological product of the groups Gp(Qp) with

respect to the distinguished subgroups Dp. We denote this group by (GA, Dp, p ∈ Rf)

or simply by (GA, Dp). That is,

(GA, Dp) = (gp) ∈∏

p∈R

Gp(Qp) | gp ∈ Dp for almost all p ∈ Rf.

As is well known, the adele group (GA, Dp) is a locally compact topological group.

If G is a connected semisimple Q-group, then we mean by (GA, G(Zp)) the adele group

attached to the groups Gp = G, p ∈ R with respect to the subgroups G(Zp), p ∈ Rf .

It is a well known result of Borel [Bo1] that the diagonal embedding of G(Q) into

(GA, G(Zp)), which we will identify with G(Q), is a lattice in (GA, G(Zp)). Furthermore

2000 Mathematics Subject Classification number: 20G35, 22E40, 22E46, 22E50, 22E55

1

2 HEE OH

Godement’s criterion in an adelic setting, proved by Mostow and Tamagawa [MT] and

also independently by Borel [Bo1], implies that G is Q-isotropic if and only if G(Q) is

a non-uniform lattice in (GA, G(Zp)).

In the spirit of Margulis arithmeticity theorem [Ma1], we show in this paper that

any irreducible lattice in an adele group (GA, Dp) is essentially of the form described as

above. We say that a lattice Γ in (GA, Dp) is irreducible if, for any finite subset S of

R containing ∞, πS (Γ ∩ (gp) ∈ GA | gp ∈ Dp for all p /∈ S) is an irreducible lattice in∏

p∈S Gp(Qp) in the usual sense (see [Ma2, Ch III, 5.9] or definition 2.9 below) where

πS denotes the natural projection (GA, Dp) →∏

p∈S Gp(Qp).

The following is a sample case of our main theorem:

1.1. Theorem. For each p ∈ R, let Gp be a connected semisimple adjoint Qp-group

without any Qp-anisotropic factors and Dp a compact open subgroup for almost all

p ∈ Rf . Assume that G∞ is absolutely simple. Then any irreducible non-uniform

lattice Γ in (GA, Dp) is rational in the sense that there exist a connected absolutely

simple Q-isotropic Q-group H and a Qp-isomorphism fp : H → Gp for each p ∈ R

with fp(H(Zp)) = Dp for almost all p ∈ Rf such that Γ is a subgroup of finite index

in f(H(Q)) where f is the restriction of the product map∏

p∈R fp to (HA, H(Zp)). In

particular, f provides a topological group isomorphism of (HA, H(Zp)) to (GA, Dp).

In order to define a rational lattice in an adele group in generality, we first de-

scribe arithmetic methods of constructing irreducible lattices in adele groups. Let

K be a number field. Let RK be the set of all (inequivalent) valuations of K. For

each v ∈ RK , Kv denotes the local field which is the completion of K with respect

to v and for non-archimedean v ∈ RK , Ov denotes the ring of integers of Kv. If

H is a connected absolutely simple K-group, it is a well known fact that the set

T (H) = v ∈ RK | H(Kv) is compact is finite. Let S be a subset of RK − T (H)

containing all archimedean valuations in RK − T (H), and let (HS ,H(Ov)) denote the

restricted topological product of the groups H(Kv), v ∈ S with respect to the subgroups

H(Ov). Then the subgroup H(K(S)), when identified with its image under the diagonal

embedding into (HS ,H(Ov)), is a lattice in (HS ,H(Ov)) where K(S) denotes the ring

of S-integers in K [Bo1]. The group H being absolutely simple, H(K(S)) is in fact an

irreducible lattice in (HS,H(Ov)).

Unless mentioned otherwise, throughout the introduction, we let Gp be a connected

semisimple adjoint Qp-group for each p ∈ R and Dp a compact open subgroup for each

p ∈ Rf .

ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 3

1.2. Definition. We call an irreducible lattice Γ in (GA, Dp) rational if there exist K,

H, S as above and a topological group epimorphism f : (GA, Dp) → (HS ,H(Ov)) with

compact kernel such that f(Γ) is commensurable with H(K(S)).

Remark.

(1) Since S ⊂ RK−T (H), H(Kv) is non-compact for each v ∈ S. If we denote by Gip

the maximal connected normal Qp-subgroup of Gp without any Qp-anisotropic

factors for each p ∈ R and let Dip = Dp ∩Gi

p for each p ∈ Rf , then in the above

definition the quotient (GA, Dp)/kerf is isomorphic to (GiA, Di

p). In particular,

if Gp(Qp) has no compact factors for any p ∈ R, we may assume that f is an

isomorphism in Definition 1.2.

(2) If R0 = p ∈ R | Gp(Qp) is non-compact, then (GiA, Di

p) is naturally identified

with the restricted topological product of the groups Gip(Qp), p ∈ R0 with respect

to the subgroups Dip. If R0 is finite, then (Gi

A, Dip) =

∏

p∈R0Gi

p(Qp). In this

case, the above definition of a rational lattice in (GA, Dp) coincides with that of

an R0-arithmetic (usually referred to as “S-arithmetic”) lattice of∏

p∈R0Gi

p(Qp)

given in [Ma2, Ch IX, 1.4].

(3) If Γ is an irreducible lattice in (GA, Dp), then pr(Γ) is an irreducible lattice in

(GiA, Di

p) as well where pr denotes the natural projection (GA, Dp) → (GiA, Di

p).

Then an irreducible lattice Γ in (GA, Dp) is rational if and only if pr(Γ) is a

rational lattice in (GiA, Di

p) in the sense of Definition A (or Definition B) in 4.1.

The following is a special case of Corollary 4.10 below.

1.3. Main Theorem. If∑

p∈R rankQp(Gp) ≥ 2, any irreducible lattice in (GA, Dp) is

rational.

That the adele group (GA, Dp) contains an irreducible lattice imposes a strong re-

striction not only on the family of the ambient groups Gp but also on the family of

distinguished subgroups Dp. The following presents a necessary and sufficient condition

on those restriction:

1.4. Theorem. For each p ∈ R, assume that Gp(Qp) has no compact factors. The

adele group (GA, Dp) admits an irreducible lattice if and only if there exist a connected

semisimple Q-simple Q-group H such that Gp is Qp-isomorphic to a connected normal

Qp-subgroup of H for each p ∈ R and Dp is a subgroup whose volume is maximum

among all compact open subgroups of Gp(Qp) for almost all p ∈ Rf .

See Theorem 4.13 below for a more general statement.

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Example.

(1) If Gp is Qp-simple and Qp-isotropic for each p ∈ R and (GA, Dp) admits an

irreducible lattice, then all Gp’s are typewise homogeneous, that is, their Dynkin

types are the same.

(2) Let n ≥ 2 and Gp = PGLn for each p ∈ R. Then (GA, Dp) has an irreducible

lattice if and only if Dp is conjugate to PGLn(Zp) for almost all p ∈ Rf .

For n = 2, for each p ∈ Rf , there are two conjugacy classes of maximal

compact open subgroups of PGL2(Qp), represented by PGL2(Zp) and by

Lp = 〈

(

a bpc d

)

∈ PGL2(Zp)

,

(

0 1p 0

)

〉

respectively. Note that in the Bruhat-Tits tree associated to PGL2(Qp), the con-

jugacy class of PGL2(Zp) corresponds to the stabilizer of a vertex and the conju-

gacy class of Lp corresponds to the stabilizer of the middle point of an edge. If we

denote by µp a Haar measure of PGL2(Qp), then µp(Lp) = 2p+1

µp(PGL2(Zp))

because the common subgroup

(

a bpc d

)

∈ PGL2(Zp)

has index p + 1 in

PGL2(Zp) while it has index 2 in Lp. Hence if Dp is conjugate to Lp for infinitely

many primes p, (GA, Dp) does not admit an irreducible lattice. Furthermore,

it follows from Theorem 1.1 and the Hasse principle that, up to automorphism

of (GA, PGL2(Zp)), PGL2(Q) is the unique irreducible non-uniform lattice in

(GA, PGL2(Zp)) up to commensurability (see Proposition 6.3 below).

(3) Let n ≥ 1. If Gp = PGSp2n for each p ∈ R, then (GA, Dp) admits an irreducible

lattice only when Dp is conjugate to PGSp2n(Zp) for almost all p ∈ Rf . Up

to automorphism of (GA, PGSp2n(Zp)), the subgroup PGSp2n(Q) is the unique

irreducible non-uniform lattice in (GA, PGSp2n(Zp)) up to commensurability

(see Proposition 6.3 below).

(4) More generally, if G is a connected absolutely simple Q-group, then for almost all

p ∈ Rf , G(Zp) is a hyperspecial subgroup of G(Qp), or equivalently, the volume

of G(Zp) is the maximum among all compact open subgroups of G(Qp) [Ti]. It

thus follows from Theorem 1.4 that if we let Gp = G, p ∈ R, then (GA, Dp)

admits an irreducible lattice if and only if Dp is conjugate to G(Zp) for almost

all p ∈ Rf (see 4.14).

If H is a connected semisimple Q-isotropic Q-group, there exists a pair P1, P2 of op-

posite proper Q-parabolic subgroups of H. Then the subgroup Ru(Pi)(Q) is a (uniform)

lattice in (HA, H(Zp)) ∩∏

p∈R Ru(Pi)(Qp) where Ru(Pi) denotes the unipotent radical


of Pi for each i = 1, 2 [Bo1]. The notation H(Q)+ denotes the subgroup generated

by all unipotent elements contained in H(Q). If H is almost Q-simple, the subgroup

generated by these two lattices Ru(P1)(Q) and Ru(P2)(Q) coincides with the subgroup

H(Q)+ [BT1]. We also show that a discrete subgroup of (GA, Dp) containing any lat-

tices in a pair of opposite horospherical subgroups respectively, is essentially of the form

H(Q)+ for H as above, under some additional assumptions on G∞.

A subgroup U of (GA, Dp) is called a horospherical subgroup if U = (GA, Dp) ∩∏

p∈R Ru(Pp) where Pp is a proper parabolic Qp-subgroups of Gp for each p ∈ R.

Two horospherical subgroups of (GA, Dp) are called opposite if the corresponding Qp-

parabolic subgroups are opposite for each p ∈ R. For a subgroup Γ ⊂ (GA, Dp), we

denote by Γ∞ the image of Γ ∩ (G∞(R) ×∏

p∈RfDp) under the natural projection

pr∞ : (GA, Dp) → G∞(R).

1.5. Theorem. For each p ∈ R, assume that Gp has no Qp-anisotropic factors. As-

sume that rank (G∞) ≥ 2. Let Γ be a subgroup of (GA, Dp) containing lattices in a pair of

opposite horospherical subgroups of (GA, Dp). Assume moreover (∗) that Γ∞ is a lattice

in G∞(R). Then Γ is discrete if and only if there exist a connected absolutely simple Q-

isotropic Q-group H and a topological group isomorphism f : (HA, H(Zp)) → (GA, Dp)

such that f(H(Q)+) ⊂ Γ ⊂ f(H(Q)).

Remark. The above theorem holds without the assumption (∗) provided Margulis’s

conjecture (see [Oh1, Conjecture 0.1]) holds for G∞. Indeed, for a discrete subgroup Γ

as above, Γ∞ is a discrete subgroup containing lattices in a pair of opposite horospherical

subgroups in G∞(R). The conjecture says that any such a discrete subgroup is a lattice

in G∞(R) as long as the real rank of G∞ is at least 2. See [Oh, Theorem 4.1] and the

remark following it for the list of groups for which the conjecture has been settled. For

instance, the list includes groups G∞ which are split over R and not locally isomorphic

to SL3(R).

We also remark that in an S-arithmetic setting (S finite), i.e., when G =∏

p∈S Gp(Qp),

the class of discrete subgroup of G containing lattices in a pair of opposite horospherical

subgroups coincides with that of non-uniform lattices in G [Oh1]. In an adelic setting

this is no more true, since the subgroup H(Q)+ has infinite index in H(Q) in general

(cf. Remark 4.12). However H(Q)+ is contained every subgroup of finite index in H(Q)

[BT1].

Naturally one may ask how many irreducible lattices an adele group (GA, Dp) can

admit up to commensurability. By the Hasse principle for an adjoint absolutely sim-

ple Q-group, Theorem 1.1 implies that for instance, if for some p ∈ R, Gp is not of

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type An (n ≥ 2), Dn (n ≥ 3), or E6, then (GA, Dp) admits at most one irreducible

non-uniform lattice up to commensurability and up to automorphism of (GA, Dp) (see

Proposition 6.3).

In section 2, we set up some notation as well as state some well known facts about

algebraic groups. In section 3, we obtain a necessary and sufficient condition on the

collection of distinguished subgroups Dp so that (GA, Dp) admits a lattice contained in

G(Q) (Theorem 3.9). Theorem 1.3 immediately follows from Theorem 4.9 (see Corollary

4.10). For the proof of Theorem 4.9, based on the S-arithmeticity theorem and a special

case of super-rigidity theorem of Margulis [Ma2], we first obtain a connected Q-simple

Q-group H so that, up to Qp-anisotropic factors, H is isomorphic to Gp, say via fp,

for each p ∈ R and H(Q) corresponds to Γ under the product map∏

p∈R fp, up to

commensurability. Here we embed Γ into∏

p∈R fp(H(Q)) ∩ (GA, Dp) as well. Up to

this part, the proof proceeds exactly the same way as in the Margulis S-arithmeticity

theorem for S finite. The difference in the case of S infinite is to handle the compact

subgroups Dp’s. To ensure the Qp-isomorphisms fp’s transfer the Q-structure on H

to (GA, Dp) in a compatible way, we has to show that fp(H(Zp)) = Dp for almost all

p ∈ Rf . This is based on the strong approximation property of the simply connected

covering of H, explained in Section 3.

Similarly the proof of Theorem 1.5 is based on the result analogous in the S-arithmetic

setting obtained by the author [Oh1]. These are explained in section 5. In section 6, we

relate the Hasse principle with the set of irreducible non-uniform lattices in (GA, Dp).

Acknowledgment. Thanks are due to Gopal Prasad and Andrei Rapinchuk for com-

munications regarding the Hasse principle for adjoint groups. I am also grateful to Jim

Cogdell, Wee Teck Gan and Dave Witte for helpful discussions.

2. Notation and Terminology

We continue the definitions and the notation mentioned in the introduction.

2.1. For any field k, we mean by a linear algebraic k-group M that M is a Zariski closed

k-subgroup of GLN for some N , and we set M(J) = M ∩GLN (J) for any subring J of

k. The term “k-group” will always mean a linear algebraic group with a fixed realization

as a k-closed subgroup of GLN for some N .

2.2. Let K be a number field. Let RK , Kv and Ov be as in the introduction. For any

S ⊂ RK , we set K(S) to be the ring of S-integers in K, that is, K(S) = x ∈ K | x ∈

Ov for all v ∈ Rf − S. If K = Q, we simply set R = RQ and we sometimes write ZS


for Q(S).

2.3. For each v ∈ RK , let Gv be a connected semisimple algebraic Kv-group. Fix a

compact open subgroup Dv of Gv(Kv) for each non-archimedean v ∈ RK . For any

subset T ⊂ RK , we denote by (GT , Dv) the restricted topological product of the groups

Gv(Kv), v ∈ T with respect to the distinguished subgroups Dv. If T is finite, then

(GT , Dv) =∏

v∈T Gv(Kv), which we will then simply denote by GT . If T = RK , we

write (GA, Dv) = (GT , Dv) and call an adele group. The topology on (GT , Dv) is given

as follows: a base of open subsets consists of the sets of the form∏

p∈S Uv ×∏

v/∈S Dv

where S is a finite subset of T and Uv ⊂ Gv(Kv) is an open subset for each v ∈ S. The

group (GT , Dv) is a locally compact group with respect to this topology.

2.4. As is well known, for each v ∈ RK , the local field Kv is a finite extension of

a subfield isomorphic to Qp for some (unique) p ∈ R. For each p ∈ R, we denote

by Ip the set of valuations v ∈ RK such that Qp ⊂ Kv (up to isomorphism). Then

Hp =∏

v∈IpRestKv/Qp

Gv is a connected semisimple Qp-group and∏

v∈IpGv(Kv) is

isomorphic to Hp(Qp) as topological groups. We denote this isomorphism by Rest0.

The map Rest0 also extends to an isomorphism of the adele group (GA, Dv) with the

adele group (HA, Mp) where Mp =∏

v∈IpRestKv/Qp

Dv (cf. [Ma2, Ch I, 3.1.4] and

[Bo1]).

2.5. If G is a connected semisimple K-group, then we mean by (GA, G(Ov)) the adele

group attached to the groups Gv = G with respect to the subgroups G(Ov). The

diagonal embedding of G(K) into (GA, G(Ov)), which we will identify with G(K), is a

lattice in (GA, G(Ov)) [Bo1]. Denote by T (G) the set of all v ∈ RK such that G(Kv) is

compact. Then T (G) is finite (cf. [Ma2, Ch I, 3.2.3]). It then follows that if T ⊂ RK

contains all archimedean valuations in RK − T (G), the subgroup G(K(T )) is a lattice

in (GT , G(Ov)) when diagonally embedded into (GT , G(Ov)). Note that

G(K(T )) = x ∈ G(K) | x ∈ G(Ov) for all non-archimedean v ∈ RK − T.

The group G is K-isotropic if and only if G(K(T )) is a non-uniform lattice in (GT , G(Ov)).

2.6. For each p ∈ R, we denote by prp the natural projection (GA, Dp) → Gp(Qp). For

any T ⊂ R, the notation prT denotes the natural projection (GA, Dp) → (GT , Dp). We

set

GA(T ) = (gp) ∈ GA | gp ∈ Dp for all p ∈ Rf − T.

For any subgroup H of (GA, Dp) and for any subset T ⊂ R, we set

HT = prT

(

H ∩ GA(T )

)

.

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2.7. For any finite S ⊂ R, let Gp be a connected semisimple algebraic Qp-group without

any Qp-anisotropic factors for each p ∈ S. A lattice Λ in GS =∏

p∈S Gp(Qp) is called

irreducible if for any two connected normal subgroups H =∏

p∈S Hp(Qp) and M =∏

p∈S Mp(Qp) of GS such that for each p ∈ S, Gp is an almost direct product of

connected normal Qp-subgroups Mp and Hp, (Λ ∩ H) · (Λ ∩ M) has infinite index in Λ

(cf. [Ma2, Ch III, 5.9]). This is equivalent to saying that Γ cannot be an almost direct

product of two infinite normal subgroups of Γ.

2.8. For a connected semisimple Qp-subgroup Gp, we denote by Gip (i standing for

isotropic) the maximal connected normal subgroup of Gp without any Qp-anisotropic

factors. Note that Gp(Qp)/Gip(Qp) is compact. That Gp has no Qp-anisotropic factors

is equivalent to saying that Gp(Qp) has no compact factors.

2.9. Let Gp be a connected semisimple algebraic Qp-group for each p ∈ R. Fix a

compact open subgroup Dp of Gp(Qp) for each p ∈ Rf . Let T ⊂ R. A lattice Γ in

(GT , Dp) is called irreducible if the projection of ΓS into∏

p∈S Gp(Qp)i is an irreducible

lattice in∏

p∈S Gp(Qp)i for any finite subset S ⊂ T containing ∞.

2.10. For a connected semisimple algebraic K-group G, the notation G(K)+ denotes

the normal subgroup of G(K) generated by the subgroups Ru(P )(K) where P runs

through the set of all parabolic K-subgroups of G and Ru(P ) denotes the unipotent

radical of P . Or equivalently G(K)+ denotes the subgroup generated by all unipotent

elements in G(K) [BT1].

If G is almost K-simple and K-isotropic, G(K)+ coincides with the subgroup gen-

erated by Ru(P1)(K) and Ru(P2)(K) for any pair P1, P2 of opposite proper parabolic

K-subgroups [BT1]. Recall that two parabolic subgroups are called opposite if their

intersection is a common Levi subgroup in both of them.

2.11. We refer to [PR], [Bo1] or [Ma2, Ch I] as a general reference to our terminology

regarding algebraic groups.

3. Distinguished subgroups Dp

3.1. Lemma. Let Gp be a connected semisimple Qp-group for each p ∈ R. Fix a

compact open subgroup Dp of Gp(Qp) for each p ∈ Rf . Let S be a finite subset of R.

Assume that ∞ ∈ S if G∞(R) is non-compact.

(1) If Γ is a discrete subgroup in (GA, Dp), then ΓS is a discrete subgroup in GS.

(2) If Γ is a (uniform) lattice in (GA, Dp), then ΓS is a (uniform) lattice in GS.


Proof. For simplicity, set GA = (GA, Dp). Denote by prS the restriction of prS to the

subgroup GA(S). Since the kernel of prS is compact, the subgroup ΓS is discrete if Γ

is so. Let Γ be a (uniform) lattice in GA. Since GA(S) is an open subgroup of GA,

the intersection Γ ∩ GA(S) is a (uniform) lattice in GA(S). Consider the natural map

prS : GA(S)/(Γ ∩ GA(S)) → GS/ΓS induced by prS . Now if µ is an invariant measure

on GA(S)/(Γ ∩ GA(S)), then prS∗ (µ) is an invariant measure on GS/ΓS. Hence if Γ is a

uniform lattice in GA, then ΓS is a uniform lattice in GS .

3.2. We recall the following corollary of (a special case of) the strong approximation

theorem:

Theorem. Let G be a connected semisimple simply connected almost Q-simple group.

Let S be a finite subset of R such that∏

p∈S Gp(Qp) is non-compact. Then G(ZS) is

dense in the direct product∏

p∈Rf−S G(Zp), when diagonally embedded. In particular

if ∆ is a subgroup of finite index in G(ZS), then the closure of ∆ has finite index in∏

p∈Rf−S G(Zp).

Proof. Let GASdenote the restricted topological product of the Gp(Qp) for p ∈ Rf − S

with respect to the subgroups G(Zp), p ∈ Rf − S.

By the strong approximation theorem (cf. [PR, Theorem 7.12, P. 427]), G(Q) is dense

in GAS. Since

∏

p∈Rf−S G(Zp) is an open subgroup of GAS, G(Q) ∩

∏

p∈Rf−S G(Zp) is

dense in∏

p∈Rf−S G(Zp). Since G(ZS) = G(Q) ∩∏

p∈Rf−S G(Zp), this proves the first

claim. For the second claim, it suffices to note that [G(ZS) : ∆] ≤ [G(ZS) : ∆] by the

first claim.

3.3. Proposition. Let p ∈ Rf . Let G be a connected semisimple Qp-group and let K1

and K2 be maximal compact subgroups of G(Qp). Assume that there exists a maximal

compact subgroup K of G(Qp) such that π(K) ⊂ K1∩K2 where G is the simply connected

covering of G and π : G → G is the Qp-isogeny. Then K1 = K2.

Proof. Consider the Bruhat-Tits building B attached to G. In the following proof, we

use some results in [Ti] without repeating reference. The group G(Qp) acts on B through

the map π. For each i = 1, 2, the maximal compact subgroup Ki is the stabilizer G(Qp)xi

in G(Qp) of some point xi ∈ B. Since G is simply connected, there exists a vertex v ∈ B

such that G(Qp)v = K. We claim that v = x1 = x2, which implies that Ki ⊂ G(Qp)

v for

both i = 1 and 2. Since K1 and K2 are maximal compact subgroups, this implies that

K1 = K2 = G(Qp)v. Suppose that v 6= xi for some i ∈ 1, 2. Since π(K) stabilizes v

and xi, it stabilizes pointwisely the unique geodesic l joining v and xi. Since v 6= xi, we

can find a facet F whose closure contains v and F ∩ l is non-empty. Fix z ∈ F ∩ l. Note

10 HEE OH

that the dimension of F is positive. Since G is simply connected, G(Qp)z = G(Qp)

F

where G(Qp)F is the pointwise stabilizer of F in G(Qp). Therefore π(K) ⊂ G(Qp)

F .

This is a contradiction, since the stabilizer of a facet of positive dimension in G(Qp)

cannot be a maximal compact subgroup. This finishes the proof.

3.4. Lemma [Ti, 3.2]. Let G be a connected semisimple Q-group. Then G(Zp) is a

maximal compact subgroup for almost all p ∈ Rf .

3.5. Lemma. Let G be a connected semisimple Q-group, G the simply connected

covering of G and π : G → G the central Q-isogeny. Then for almost all p ∈ Rf ,

π(G(Zp)) ⊂ G(Zp).

See [PR, P. 451] for the proof of Lemma 3.5.

3.6. Proposition. Let G, G and π be as in Lemma 3.5. For each p ∈ Rf , let Dp be a

compact open subgroup of G(Qp). Assume that π(G(Zp)) ⊂ Dp for almost all p ∈ Rf .

Then Dp ⊂ G(Zp) for almost all p ∈ Rf .

Proof. Since every compact open subgroup of G(Qp) is contained in a maximal compact

open subgroup, we may assume that Dp is a maximal compact open subgroup for

all p ∈ Rf . By Lemmas 3.4 and 3.5, there exists a finite subset S ⊂ Rf such that

for each p ∈ Rf − S, both Dp and G(Zp) are maximal compact subgroups of G(Qp)

and π(G(Zp)) ⊂ Dp ∩ G(Zp). Applying Proposition 3.3, we have Dp = G(Zp) for all

p ∈ Rf − S, proving the claim.

3.7. The following is a special case of [Ma2, Ch IX, 4.15].

Theorem. Let G be a connected semisimple almost Q-simple Q-group. Let S be a

finite subset of R such that∑

p∈S rank QpG ≥ 2. If G(R) is non-compact, we assume

that ∞ ∈ S. If Γ ⊂ G(Q) and Γ is a lattice in GS (when diagonally embedded), then

Γ and G(ZS) are commensurable. In particular Γ contains a subgroup of finite index in

G(ZS).

3.8. Set G(Q) ∩ (GA, Dp) = x ∈ G(Q) | x ∈ Dp for almost all p ∈ Rf. We identify

this set with its image under the diagonal embedding into (GA, Dp).

Theorem. Let G and S be as in Lemma 3.7. Let Dp be a compact open subgroup of

G(Qp) for each p ∈ Rf . If Γ is a closed subgroup of G(Q) ∩ (GA, Dp) such that ΓS is

a lattice in GS =∏

p∈S G(Qp), then π(G(Zp)) ⊂ Dp ⊂ G(Zp) for almost all p ∈ Rf ,

where G and π are as in Lemma 3.5. Furthermore if Γ is a lattice in (GA, Dp), then

Dp = G(Zp) for almost all p ∈ Rf .


Proof. We first consider the case when G is simply connected. By Theorem 3.7, ΓS

contains a subgroup of finite index in G(ZS). Let ∆ denote the diagonal embedding of

the subgroup ΓS∩G(ZS) into∏

p∈Rf−S G(Zp). Note that ∆ ⊂∏

p∈Rf−S(G(Zp)∩Dp) ⊂∏

p∈Rf−S G(Zp). By Theorem 3.2, the closure ∆ of ∆ is a compact open subgroup in∏

p∈Rf−S G(Zp), and hence G(Zp) ⊂ prp(∆) for almost all p ∈ Rf . On the other

hand, since∏

p∈Rf−S(G(Zp) ∩ Dp) is compact, we have prp(∆) ⊂ G(Zp) ∩ Dp for each

p ∈ Rf − S. Therefore G(Zp) = Dp for almost all p ∈ Rf by Lemma 3.4.

If G is not simply connected, consider the simply connected covering G and the Q-

isogeny π : G → G. Denote by πp the restriction of π to G(Qp). By Lemma 3.5, we have

that G(Zp) = π−1p G(Zp) for almost all p ∈ Rf . Set Γ = G(Q)∩(GA, π−1

p (Dp)). Since the

kernel of∏

p∈S πp is finite, it is clear that ΓS is a lattice in∏

p∈S G(Qp). Therefore, by

the previous simply connected case, we have π−1p (Dp) = G(Zp); hence πp(G(Zp)) ⊂ Dp

for almost all p ∈ Rf . Therefore for almost p ∈ Rf , πp(G(Zp)) ⊂ Dp ⊂ G(Zp) by

Proposition 3.6.

Now assume that Γ is a lattice in (GA, Dp). Recall that G(Q) is a lattice in (GA, G(Zp)).

Denote by µ1 and µ2 the Haar measures on (GA, Dp) and (GA, G(Zp)) normalized so

that µ1(Dp) = 1 and µ(G(Zp)) = 1 for all p ∈ Rf , respectively. For each finite S ⊂ R

containing ∞, ΓS is a subgroup of finite index in G(ZS) and both are lattices in GS .

Hence µ2(GS/ΓS) ≥ µ2(GS/G(ZS)). Note that

µ2(GS/ΓS) =µ1(GS/ΓS)

∏

p∈Rf∩S [G(Zp) : Dp].

For any increasing sequence Si, i = 1, 2, · · · such that R = ∪iSi, the measures µ2(GSi/G(ZSi

))

and µ1(GSi/ΓSi) converge to the measures µ2((GA, G(Zp))/G(Q)) and µ1((GA, Dp)/Γ)

respectively. Hence limi→∞

∏

p∈Rf∩Si[G(Zp) : Dp] should be bounded. Hence Dp =

G(Zp) for almost all p ∈ Rf .

3.9. Theorem. Let G be a connected semisimple almost Q-simple group and Dp a

compact open subgroup of G(Qp) for each p ∈ Rf . If G is not simply connected, assume

that Dp is a maximal compact open subgroup for almost all p ∈ Rf . Then the following

are equivalent.

(1) The adele group (GA, Dp) admits a lattice contained in G(Q) ∩ (GA, Dp).

(2) There exists a finite subset S ⊂ R such that∑

p∈S rank QpG ≥ 2, and ΓS is a

lattice in∏

p∈S G(Qp) where Γ = G(Q) ∩ (GA, Dp).

(3) Dp = G(Zp) for almost all p ∈ Rf .

12 HEE OH

Proof. To show (1) ⇒ (2), since G(Qp) is non-compact for almost all p ∈ R (see 2.5),

there exists a finite subset S ⊂ R such that∑

p∈S rank QpG ≥ 2. If G∞(R) is non-

compact, we assume ∞ ∈ S. Let ∆ be a lattice in (GA, Dp) contained in G(Q). It is not

difficult to check that Γ is a discrete subgroup of (GA, Dp). Since ∆ ⊂ Γ, the subgroup Γ

is a lattice in (GA, Dp) as well. Hence ΓS is a lattice in∏

p∈S G(Qp) by Lemma 3.1. The

direction (2) ⇒ (3) follows from Theorem 3.8. If (3) holds, then (GA, Dp) = (GA, G(Zp))

in the sense that the identity map provides a topological group isomorphism between

them, and hence G(Q) is a lattice in (GA, Dp).

3.10. The second claim in Theorem 3.8 combined with Theorem 3.9 yields the following:

Theorem. Let G be a connected semisimple almost Q-simple group and Dp a compact

open subgroup of G(Qp) for each p ∈ Rf . Then the following are equivalent.

(1) The adele group (GA, Dp) admits a lattice contained in G(Q) ∩ (GA, Dp).

(2) Dp = G(Zp) for almost all p ∈ Rf .

4. Rationality of an irreducible lattice in GA

4.1. In the following we give two definitions of a rational lattice. It is convenient for

our purpose to understand the equivalence of the two definitions.

In both definitions, let T ⊂ R and let Gp be a connected semisimple adjoint Qp-group

without any Qp-anisotropic factors for each p ∈ T and Dp a compact open subgroup of

G(Qp) for almost all finite p ∈ T .

Definition A. An irreducible lattice Γ in (GT , Dp) is called a rational lattice if there

exist:

(1) a connected semisimple adjoint Q-simple Q-group H;

(2) if ∞ /∈ T , H(R) is compact;

(3) for each p ∈ T , a decomposition H = H1p ×H2

p where H1p and H2

p are connected

semisimple adjoint Qp-groups;

(4) for each p ∈ T , a maximal compact open subgroup Mp ⊂ H2p(Qp) with Mp =

H(Zp) ∩ H2p (Qp) for almost all finite p ∈ T ; and

(5) a family of Qp-epimorphisms fp : H → Gp, p ∈ T with kerfp = H2p and

fp(H(Zp)) = Dp for almost all finite p ∈ T

such that Γ is commensurable with the subgroup

f

H(Q(T )) ∩∏

p∈T

(H1p(Qp) × Mp)


where f =∏

p∈T fp.

Definition B. An irreducible lattice in (GT , Dp) is called a rational lattice if there

exist a number field K, a connected absolutely simple K-group H and a subset B ⊂

RK − T (H) containing all archimedean valuations in RK − T (H) such that there ex-

ists a topological group isomorphism f : (HB,H(Ov)) → (GT , Dp) and f(H(K(B)) is

commensurable with Γ.

Remark. When T is finite, Definition A (essentially) coincides with that of an S-

arithmetic lattice given in [Zi, Theorem 10.1.12] and Definition B coincides with that of

an S-arithmetic lattice given in [Ma2, Ch IX]

4.2. Proposition. Definitions A and B are equivalent.

Proof. Assume that Γ is a rational lattice in (GT , Dp) as in Definition A. For any H

as in (1), there exist a number field K and a connected absolutely simple K-group H

such that H = RestK/Q H. For each p ∈ T , let Ip ⊂ RK be as in 2.4. Then considering

H as a Qp-group and H as a Kv-group for each v ∈ Ip, we have the decomposition

H =∏

v∈IpRestKv/Qp

H over Qp. Since H is a connected absolutely simple Kv-group,

each RestKv/QpH is a connected semisimple adjoint Qp-simple Qp-group. For each

p ∈ T , we can find a partition of Ip into I1p ∪ I2

p such that H1p =

∏

v∈I1pRestKv/Qp

H

and H2p =

∏

v∈I2pRestKv/Qp

H = kerfp. Set B = ∪p∈T I1p . Note that if ∞ ∈ T , H2

∞(R)

is compact, since otherwise H2∞(R) does not admit a compact open subgroup. Hence if

v ∈ RK is an archimedean valuation with H(Kv) non-compact, then v /∈ I2∞. That is,

I1∞ and hence B contains all archimedean valuations in RK −T (H). If ∞ /∈ T , H(R) is

compact, and hence for all archimedean v ∈ I∞, H(Kv) is compact, that is, v ∈ T (H).

Since Gp(Qp) has no compact factors, B ⊂ RK − T (H).

Since Mp = H(Zp) ∩ H2p (Qp) for almost all finite p ∈ T and M∞ has finite index

in H2∞(R) in the case when ∞ ∈ T , the subgroup RestK/Q H(K(B)) is commensurable

with

x ∈ H(Q(T )) | x ∈ H1p(Qp) × Mp for each p ∈ T.

Via the map Rest0 (see 2.4), the group (HB ,H(Ov)) is isomorphic to (H1T , H(Zp) ∩

H1(Qp)) and the lattice H(K(B)) is mapped to a subgroup commensurable with the

subgroup

pr1 δT

(

x ∈ H(Q(T )) | x ∈ H1p (Qp) × Mp for each p ∈ T

)

where δT denotes the diagonal embedding of H(Q) into (HT , H(Zp)) and pr1 denotes

the canonical projection (HT , H(Zp)) → (H1T , H(Zp) ∩ H1

p(Qp)).

14 HEE OH

Note that Rest0(∏

v∈I1pH(Ov)) = H(Zp) ∩ H1

p (Qp) for each finite p ∈ T . Hence

if f1 denotes the restriction of the map f to (H1T , H(Zp) ∩ H1(Qp)), then f1 Rest0

is an isomorphism of (HB,H(Ov)) to (GT , Dp) and the image of H(K(B)) under this

isomorphism is commensurable with Γ. Hence Γ is a rational lattice in Definition B as

well.

To see the converse, if we let H = RestK/Q H, then H is a connected semisimple ad-

joint Q-simple Q-group. Let T = p ∈ R | for some v ∈ B, Kv is a finite extension of Qp.

Note that H(R) is non-compact if and only if H(Kv) is non-compact for an archimedean

valuation v ∈ RK . Hence if H(R) is non-compact, then ∞ ∈ T , since B contains

all archimedean valuations in RK − T (H). Let I1p = Ip ∩ B and I2

p = Ip − I1p .

Set H1p =

∏

v∈I1pRestKv/Qp

H and H2p =

∏

v∈I2pRestKv/Qp

H. Note that if ∞ ∈ T ,

H2∞(R) is compact. It follows from the lemma below that there exist Qp-isomorphisms

hp : H1p → Gp, p ∈ T such that hp(H

1p (Qp) ∩ Hp(Zp)) = Dp for almost all finite p ∈ T

and f =∏

p∈T hp Rest0 where f : (HB,H(Ov)) → (GT , Dp) is the given topologi-

cal group isomorphism and Rest0 :∏

v∈I1pH(Kv) → H1

p(Qp) as in 2.4. If prp denotes

the natural projection H → H1p , then the map fp = hp prp is a Qp-epimorphism

from H → Gp with kerfp = H2p and fp(H(Zp)) = Dp for almost all finite p ∈ T . Set

Mp = H(Zp) ∩ H2p (Qp) for each finite p ∈ T . If ∞ ∈ T , set M∞ = H2

∞(R). Then

H(K(B)), is commensurable to the subgroup

x ∈ H(K(B0)) |∏

v∈Ip

RestKv/Qpx ∈ H1

p × Mp for each p ∈ T

where B0 = ∪p∈T Ip. Therefore via the map∏

p∈T fp, Γ is commensurable to

x ∈ H(Q(T )) | x ∈ H1p(Qp) × Mp for each p ∈ T.

Hence Γ is rational as in Definition A.

We formulate the lemma used in the above proof.

4.3. Lemma. Let S, T ⊂ R. Let Gp, p ∈ S (resp. Hp, p ∈ T ) be connected semisimple

adjoint Qp-groups without any Qp-anisotropic factors and Mp ⊂ Gp(Qp) (resp. Lp ⊂

Hp(Qp)) compact open subgroups for each finite prime p ∈ S (resp. p ∈ T ). Assume

that Mp and Lp are maximal compact subgroups for almost all finite p ∈ S. If f :

(GS , Mp) → (HT , Lp) is a topological group isomorphism, then S = T , there exist Qp-

isomorphisms fp : Gp → Hp, p ∈ S such that fp(Mp) = Lp for almost all finite p ∈ S

and f(x) =∏

p∈S fp(x) for any x ∈ (GS , Mp).


Proof. For each p ∈ S and r ∈ T , consider the map fpr : Gp(Qp) → Hr(Qr) de-

fined by fpr(g) = prr(f(g)) for each g ∈ Gp(Qp). Here prr : (HT , Lp) → Hr(Qr)

denotes the natural projection map. Then fpr is a continuous homomorphism for each

p and r. Since f is an isomorphism, for each p ∈ S, fpr(Gr(Qr)) 6= e for some

r ∈ T . By [Ma2, Ch I, Proposition 2.6.1], we have fpr(Gr(Qr)) 6= e if and only if

p = r, and when p = r, the topological group isomorphism fpp : Gp(Qp) → Hp(Qp)

extends to a rational Qp-isomorphism fp : Gp → Hp. It follows that S = T . Since

the restriction of f to (GS∩Rf, Mp) induces an isomorphism f ′ : (GS∩Rf

, Mp) →

(HT∩Rf, Lp), the image f ′(

∏

p∈S∩RfMp) is an open compact subgroup of (HT∩Rf

, Lp).

Since f ′(∏

p∈S∩RfMp) =

∏

p∈S∩Rffp(Mp) and fp(Mp) ⊂ Hp(Qp) for each p ∈ S ∩ Rf ,

Lp ⊂ fp(Mp) for almost all finite p ∈ S. Since Mp and Lp are maximal compact for

almost all finite p ∈ S, we have that Lp = fp(Mp) for almost all finite p ∈ S.

4.4. Margulis’s S-arithmeticity theorem states:

Theorem. Let S ⊂ R be a finite subset and let Gp be a connected semisimple adjoint

Qp-group without any Qp-anisotropic factors for each p ∈ S. If∑

p∈S rank Qp(Gp) ≥ 2,

any irreducible lattice in GS is an S-arithmetic lattice in GS.

See [Ma2, Ch IX, Theorem 1.11 and the remark 1.3. (iii)] or [Zi, Theorem 10.1.12].

4.5. Before we give a proof of rationality theorem which works for uniform and non-

uniform lattices simultaneously, we give an instructive simpler proof for an irreducible

non-uniform lattice assuming that G∞ is absolutely simple. Theorem 1.1 immediately

follows from the following:

Theorem. For each p ∈ R, let Gp be a connected semisimple adjoint Qp-group without

any Qp-anisotropic factors. For each p ∈ Rf , let Dp ⊂ Gp(Qp) be a compact open

subgroup. Assume that G∞ is absolutely simple. Fix a finite subset S0 ⊂ R containing

∞ such that∑

p∈S0rank Qp

(Gp) ≥ 2. Let Γ be a subgroup of (GA, Dp) such that ΓS

is an irreducible non-uniform lattice in GS for any finite S ⊂ T including S0. Then

there exist a connected absolutely simple Q-isotropic Q-group H and a Qp-isomorphism

fp : H → Gp for each p ∈ R with fp(H(Zp)) = Dp for almost all p ∈ Rf such that

Γ ⊂ f(H(Q)) where f is the restriction of∏

p∈R fp to (HA, H(Zp)).

Proof. Set Ω = S ⊂ R | S0 ⊂ S, |S| < ∞.

Step 1. Obtain Q-forms HS for each S ∈ Ω. For any S ∈ Ω, by Theorem 4.4 and

Definition A, there exist a connected absolutely simple Q-group HS , Qp-isomorphisms

16 HEE OH

rSp : HS → Gp, p ∈ S such that ΓS is commensurable with the subgroup

(rSp(x)) | x ∈ HS(ZS).

Since the groups Gp, p ∈ S and HS are adjoint, we may assume that ΓS is a finite index

subgroup of (rSp(x)) | x ∈ HS(ZS).

Step 2. The Q-forms HS are all Q-isomorphic. Set H = HS0∞ and r = rS0∞. By

the assumption and Lemma 3.1, Γ∞ is a lattice in G∞(R) and hence is Zariski dense in

G∞ by Borel density theorem. For any S ∈ Ω, since Γ∞ ⊂ rS∞(HS(Q))∩ r(H(Q)) and

r r−1S,∞(Γ∞) ⊂ HS(Q), the map r r−1

S∞ : H → HS is defined over Q [Ma2, Ch I, 0.11].

Since both H and HS are absolutely simple, the map rr−1S∞ is indeed a Q-isomorphism.

Step 3. Define a Qp-isomorphism fp : H → Gp for each p ∈ R. For each p ∈ R, we

define a map fp : H → Gp by fp = rSp (rS∞)−1 r for any S ∈ Ω containing p. To

show that this is independent of the choice of S, we claim that for any p ∈ R and for

any S1, S2 ∈ Ω such that p ∈ S1 ∩ S2, rS1p r−1S1∞

= rS2p r−1S2∞

. Since

Γ∞,p ⊂ (rS1∞(x), rS1p(x)) | x ∈ HS1(Q) ∩ (rS2∞(x), rS2p(x)) | x ∈ HS2

(Q),

we have that rS1pr−1S1∞

(z) = rS2pr−1S2∞

(z) for any z ∈ pr∞(Γ∞,p). Since pr∞(Γ∞,p)

is a Zariski dense subset in G∞, rS1p r−1S1∞

= rS2p r−1S2∞

and hence the map fp is

well defined for each p ∈ R. Since rSp is a Qp-isomorphism and (rS∞)−1 r is a

Q-isomorphism, fp is a Qp-isomorphism.

Step 4. Show Γ ⊂ f(H(Q)) where f =∏

p∈R fp . We now claim that Γ ⊂ f(H(Q))

where f =∏

p∈R fp and f(H(Q)) = (fp(x)) | x ∈ H(Q). It suffices to show that ΓS ⊂

fS(H(Q)) = (fp(x))p∈S | x ∈ H(Q) for each S ∈ Ω. But ΓS ⊂ (rSp(x))p∈S | x ∈

HS(Q). If x ∈ HS(Q), then there exists a unique y ∈ H(Q) such that x = r−1S∞ r(y).

Hence rSp(x) = fp(y) for each p ∈ S. Therefore ΓS ⊂ fS(H(Q)) for any S ∈ Ω.

Step 5. Show fp(H(Zp)) = Dp for almost all p ∈ Rf . The product map f induces a

topological group isomorphism from (HA, f−1p (Dp)) to (GA, Dp). Note that f−1(Γ) ⊂

H(Q) ∩ (HA, f−1p (Dp)). Since f−1(Γ)S is a lattice in

∏

p∈S H(Qp) for any S ∈ Ω, by

Theorem 3.10, we have f−1p (Dp) = H(Zp), or equivalently fp(H(Zp)) = Dp, for almost

all p ∈ Rf . This finishes the proof.

4.6. The proof of Theorem 1.3 is more involved in general cases. We will need the

following preparation before giving its proof. In view of the equivalence of the two

definitions given in 4.1, the following is a direct corollary of [Ma2, Ch VIII, Theorem

3.6]:


Proposition. Let H be a connected semisimple adjoint Q-simple Q-group. Let S be a

finite subset of R. Assume that ∞ ∈ S if H(R) is non-compact. For each p ∈ S, let

H = H1p × H2

p where the subgroups H1p and H2

p are connected normal Qp-subgroups of

H, H1p has no Qp-anisotropic factors and Mp ⊂ H2

p(Qp) a compact open subgroup. Let

F be a connected adjoint semisimple Q-group, Λ a subgroup of H commensurable with

x ∈ H(Q(S)) | x ∈ (H1p (Qp)×Mp) for each p ∈ S and δ : Λ → F (Q) a homomorphism

with a Zariski dense image in F . Assume that∑

p∈S rankQp(H1

p ) ≥ 2. Then there exists

a (unique) Q-isomorphism j : H → F which extends δ.

4.7. Lemma. Let S and Gp be as on Theorem 4.4. If Γ is an irreducible lattice in GS.

Then for any p ∈ S, the restriction of prp : GS → Gp to Γ is injective.

Proof. If S = p, the statement is trivial. Suppose not. Set N = γ ∈ Γ | prp(γ) = e.

Then N is a normal subgroup of Γ. Since the lattice Γ is irreducible, the image of Γ

under prp is infinite. Hence N is not commensurable with Γ. By Margulis’s normal

subgroup theorem [Ma2, Ch VIII, Theorem 2.6], N is contained in the center of GS .

Since the groups Gp are adjoint, the center of GS is trivial, proving the claim.

4.8. Lemma. For any p ∈ R, let G be a connected reductive Qp-group. Then any

compact open subgroup of Gp(Qp) is contained in only a finitely number of compact

subgroups of Gp(Qp).

Proof. If G(R) contains a compact open subgroup, say U , it follows that G(R) itself is

compact and U has a finite index in G(R). Hence the claim follows. For a finite prime

p, see [PR, Proposition 3.6, P 136].

4.9. We are now ready to prove the main theorem:

Theorem. Let T ⊂ R. For each p ∈ T , let Gp be a connected semisimple adjoint Qp-

group without any Qp-anisotropic factors. For almost all finite p ∈ T , let Dp ⊂ Gp(Qp)

be a maximal compact open subgroup. Fix a finite subset S0 ⊂ T (containing ∞ if

∞ ∈ T ) such that∑

p∈S0rank Qp

(Gp) ≥ 2. Let Γ be a subgroup of (GT , Dp) such that

ΓS is an irreducible lattice in GS for any finite S ⊂ T including S0. Then Γ is contained

in some rational lattice in (GT , Dp).

Proof. Set p0 = ∞ if ∞ ∈ T , and otherwise let p0 be any fixed prime in S0. Set

Ω = S ⊂ T | S0 ⊂ S, |S| < ∞.

Step 1. Obtain the Q-forms HS, S ∈ Ω. For each S ∈ Ω, we denote by HS , H1Sp,

H2Sp, MSp, fSp, prSp as in Theorem 4.4 and Definition A in 4.1. Also set rSp to be the

18 HEE OH

composition map fSpprSp : HS → Gp. Since the groups Gp, p ∈ S and HS are adjoint,

we may assume that ΓS is a subgroup of finite index in

(rSp(x))p∈S | x ∈ HS(ZS) ∩ (H1Sp × MSp) for each p ∈ S.

Step 2. The Q-forms HS are all Q-isomorphic. Set H = HS0and r = rS0p0

. We

first claim that the group HS is Q-isomorphic to H for any S ∈ Ω. Consider the maps

r : H → Gp0and rSp0

: HS → G∞. For simplicity, we set prp0= pr0. Since pr0(Γ

S) ⊂

rSp0(HS(ZS)) for any S ∈ Ω, the set r−1

Sp0(pr0(Γ

S)) = x ∈ HS(ZS) | r(x) ∈ prSp0(ΓS)

is contained in HS(ZS).

Since the map rSp0is injective over r−1

Sp0(pr0(Γ

S0)) by Lemma 4.7, the composition

map r−1Sp0

r is well defined on r−1(pr0(ΓS0)), which we denote by jSp0

: r−1(pr0(ΓS0)) →

HS(Q). By Proposition 4.6, the map jSp0extends to a Q-rational isomorphism jS : H →

HS , proving our claim.

Step 3. Define Qp-epimorphisms fp : H → Gp. For each p ∈ T , we define a map

fp : H → Gp by fp = rSp jS for any S ∈ Ω containing p. To show that this is

independent of the choice of S, we claim that for any p ∈ T and for any S1, S2 ∈ Ω

such that p ∈ S1 ∩ S2, rS1p jS1= rS2p jS2

. Note that r is injective over H(ZS0) by

Lemma 4.7. We let r−1(ΓS0) = x ∈ H(ZS0) | r(x) ∈ ΓS0.

Since HS0is Q-simple, it follows that r−1(pr0(Γ

S0)) is Zariski dense in HS0. Hence

it suffices to verify this equality for any x ∈ r−1(pr0(ΓS0)). There exists a unique

(see Lemma 4.7) element y ∈ ΓS0 such that r(x) = pr0(y), and there exist elements

zi ∈ HSi(Q), i = 1, 2 such that y = prS0

(rS1q(z1)) = prS0(rS2q(z2)). Again by Lemma

4.7, we have rS1p(z1) = rS2p(z2). Then rS1p jS1(x) = rS1p r−1

S1p0 r(x) = rS1p(z1)

which is equal to rS2p(z2) = rS2p r−1S2p0

r(x) = rS2p jS2(x). This proves our claim,

yielding that fp is well defined for each p ∈ T . Since rSp is a Qp-epimorphism and jS is

a Q-isomorphism, fp is a Qp-epimorphism.

Step 4. The groups H1p , H2

p and Mp. Note that kerfp is a connected semisimple

adjoint Qp-group, as is any connected normal Qp-subgroup of H. Letting H2p = kerfp

for each p ∈ T , there exists a connected semisimple adjoint Qp-group H1p such that

H = H1p × H2

p . Note that the restriction fp : H1p → Gp is a Qp-isomorphism. Since jS

is a Q-isomorphism, H1p = j−1

S (H1Sp) and H2

p = j−1S (H2

Sp) for each S ∈ Ω containing p.

We claim that there exists a compact open subgroup, say, Mp, of H2p(Qp) such

that f−1p (prp(Γ

S)) ⊂ Mp for each S ∈ Ω containing p. Note that f−1p (prp(Γ

S0)) ⊂

f−1p (prp(Γ

S)) for each S ∈ Ω containing p. On the other hand, the latter subgroup is


contained in the compact open subgroup j−1S (MSp) ⊂ H2

Sp(Qp). Since f−1p (prp(Γ

S0)) ⊂

∩S∈Ωj−1S (MSp), ∩S∈Ωj−1

S (MSp) is a compact open subgroup of H2p(Qp). Hence by

Lemma 4.8, ∪p∈S∈Ωj−1S (MSp) = j−1

Sm(MSmp) for some Sm ∈ Ω. It suffices to set Mp to

be a maximal compact open subgroup of H2p(Qp) containing j−1

Sm(MSmp).

Step 5. Show Γ ⊂ f(H(Q)) where f =∏

p∈T fp . We now claim that Γ ⊂ f(H(Q) ∩∏

p∈T (H1p × Mp)) where f =

∏

p∈T fp. It suffices to show that ΓS ⊂ fS(H(Q) ∩∏

p∈R(H1p × Mp)) for each S ∈ Ω, where fS =

∏

p∈S fp. For any γ ∈ ΓS , we have

γ = (rSp(x))p∈S for some x ∈ HS(Q) and in the decomposition x = x1px

2p for x1

p ∈ H1Sp

and H2Sp, we have x2

p ∈ Mp. Since x ∈ HS(Q), then there exists a unique y ∈ H(Q) such

that x = jS(y). Hence rSp(x) = fp(y) and y = j−1S (x1

p)j−1S (x2

p) where j−1S (x1

p) ∈ H1p and

j−1S (x2

p) ∈ j−1S (ker prSp) ⊂ Mp for each p ∈ S. Therefore ΓS ⊂ fS(H(Q) ∩

∏

p∈R(H1p ×

Mp)) for any S ∈ Ω.

Step 6. fp(H(Zp)) = Dp for almost all finite p ∈ T . For each p ∈ T , set D′p = x ∈

H1p (Qp) | fp(x) ∈ Dp. For p ∈ T , set Lp = D′

p × Mp and for p ∈ Rf − T , Lp = H(Zp).

Consider the adele group (HA, Lp). Now the subgroup x ∈ H(Q) | fT (x) ∈ Γ satisfies

the property (2) in Theorem 3.8 where fT =∏

p∈T fp. Hence Lp = H(Zp) for almost

all p ∈ Rf , and hence we have D′p ×Mp = H(Zp) and Mp = H(Zp)∩H2

p (Qp) for almost

all finite p ∈ T . Therefore fp(H(Zp)) = Dp for almost all finite p ∈ T . Therefore we

have constructed (H, fp, Mp) as required in Definition A.

4.10. Corollary. Let T ⊂ R. For each p ∈ T , let Gp be a connected semisimple adjoint

Qp-group without any Qp-anisotropic factors. For each finite p ∈ T , let Dp ⊂ Gp(Qp)

be a compact open subgroup. If∑

p∈T rank Qp(Gp) ≥ 2. then any irreducible lattice in

(GT , Dp) is rational.

Proof. The condition on maximality of Dp’s was used only in Step 6 in the above proof.

Here instead of referring to Theorem 3.9, it suffices to refer to Theorem 3.10 to deduce

Lp = H(Zp) for almost all p ∈ Rf . Then the rest proceeds exactly the same way.

4.11. Without the assumption of Gp being adjoint, we can deduce the following from

Theorem 4.5 and Theorem 3.9:

Proposition. For each p ∈ R, let Gp be a connected semisimple Qp-group without any

Qp-anisotropic factors and let G∞ be absolutely almost simple. Let Dp be a compact

open subgroup of Gp(Qp) for each p ∈ Rf . If Γ is an irreducible non-uniform lattice in

(GA, Dp), then there exists a connected absolutely simple Q-group H and a Qp-isogeny

fp : Gp → H for each p ∈ R such that π(H(Zp)) ⊂ fp(Dp) ⊂ H(Zp) for almost p ∈ R

20 HEE OH

and∏

p∈R fp(Γ) ⊂ H(Q) where H is the simply connected covering of H and π : H → H

is the Q-isogeny.

Example. Let n ≥ 2 and Gp = SLn for each p ∈ R. Let Dp be a (not necessarily

maximal) compact open subgroup of SLn(Qp) for each p ∈ R. If (GA, Dp) has an non-

uniform irreducible lattice, then for almost all p ∈ Rf , Dp is conjugate to SLn(Zp) by

an element of GLn(Qp).

4.12. Remark. We remark that the subgroup Γ need not be a lattice in GA to satisfy the

assumptions in Theorem 4.5 or 4.9. Let G be a connected absolutely simple Q-isotropic

Q-group. If G(Q)+ ⊂ Λ ⊂ G(Q), ΛS is an irreducible lattice in GS for any finite set

S containing ∞ such that∑

p∈S rank Qp(Gp) ≥ 2. Indeed, ΛS is a discrete subgroup

of GS such that G(Q)+S⊂ ΛS ⊂ G(Q)

S. Note that G(ZS) = G(Q)

Sand G(Q)+

Sis

an infinite normal subgroup of G(ZS). Hence by Margulis’s normal subgroup theorem,

G(Q)+S

has finite index in G(ZS) ([Ma2, Ch VIII, Theorem 2.6]). Therefore ΛS is a

lattice in GS . From the assumption that G is absolutely simple, the subgroup G(ZS)

and hence ΛS is an irreducible lattices in GS .

However G(Q)+ does not have finite index in G(Q) in general. If we denote by G the

simply connected covering of G and π : G → G is the Q-isogeny, then G(Q)+ = π(G(Q)).

Suppose that H1(Q, G) is trivial, this happens for example if G = SLn. Let C denote

the kernel of π. From the exact sequence 1 → C → G → G → 1 it follows that

G(Q)/ G(Q)+ = G(Q)/π(G(Q)) ≈ H1(Q, C).

If G = PGLn and G = SLn, then

H1(Q, C) = H1(Q, µn) = Q∗/(Q∗)n

where µn is the n-th root of unity.

4.13. For a connected semisimple Q-group H, for almost all p ∈ Rf , H is unramified

over Qp, that is, quasi-split over Qp and split over an unramified extension of Qp. For

such primes p ∈ Rf , H(Zp) is a hyperspecial subgroup of H(Qp) or equivalently, a

compact subgroup whose volume is maximum among all compact subgroups of H(Qp).

Hyperspecial subgroups of H(Qp) are conjugate to each other by an element of Had(Qp)

where Had is the adjoint group of H [Ti, 3.8].

Theorem. Let T ⊂ R and let Gp be a connected semisimple adjoint Qp-group without

any Qp-anisotropic factors for each p ∈ T . Assume that∑

p∈T rank Qp(Gp) ≥ 2. Then


the group (GT , Dp) admits an irreducible lattice if and only if there exist a connected

semisimple Q-simple Q-group H such that Gp is Qp-isomorphic to a connected normal

Qp-subgroup of H for each p ∈ T and Dp is a subgroup whose volume is maximum

among all compact subgroups of Gp(Qp) for almost all finite p ∈ T .

Proof. The “only if” direction follows from Corollary 4.10, Definition A in 4.1 and the

above remark. To see the other direction, denote by fp : H → Gp a Qp-epimorphism

for each p ∈ T . Let S be a finite subset of Rf such that for any p ∈ Rf − S, H

unramified over Qp and H(Zp) is a hyperspecial subgroup of H(Qp). By the hypothesis

on Dp, we can find a hyperspecial subgroup D′p ⊂ H(Qp) such that Dp ⊂ f(D′

p) for each

p ∈ (Rf −S)∩T . Hence for each p ∈ (Rf −S)∩T , there exists gp ∈ Had(Qp) such that

gpD′pg

−1p = H(Zp). Let φp = intgp if p ∈ (Rf − S) ∩ T and φp = id if p ∈ R ∩ S,∞∩

T . Then φ =∏

p∈T φp yields a topological group isomorphism between (HT , D′p) and

(HT , H(Zp)). Since H(Q(T )) is an irreducible (H being almost Q-simple) lattice in

(HT , H(Zp)), φ−1(H(Q(T ))) is an irreducible lattice in (HT , D′p). For each p ∈ T , write

H = H1p ×kerfp and D′

p = M1p ×M2

p so that M1p ⊂ H1

p (Qp) and M2p ⊂ kerfp(Qp). Since

∏

p∈T (H1p(Qp) × M2

p ) ∩ (HT , D′p) is an open subgroup of (HT , D′

p), the intersection

φ−1(H(Q)) ∩∏

p∈T (H1p (Qp) × M2

p ) is a lattice in∏

p∈T (H1p(Qp) × M2

p ) ∩ (HT , D′p).

Now the canonical projection pr :∏

p∈T (H1p (Qp) × M2

p ) ∩ (HT , D′p) → (H1

T , M1p ) has

compact kernel, the subgroup pr(φ−1(H(Q)) ∩∏

p∈T (H1p (Qp) × M2

p )) is a lattice in

(H1T , M1

p ). Since the restriction of∏

p∈T fp provides a topological group isomorphism

from (H1T , M1

p ) onto (GT , Dp), we obtain a lattice in (GT , Dp). Since H is Q-simple, the

lattice obtained this way is irreducible, otherwise, it would yield a proper Q-subgroup

of H.

4.14. Corollary. Let H be a connected absolutely simple Q-group. Let Dp ⊂ H(Qp)

be a compact open subgroup for each p ∈ Rf . If (HA, Dp) admits an irreducible lattice,

then Dp is conjugate to H(Zp) for almost all p ∈ Rf .

5. Discrete subgroups containing lattices in horospherical subgroups

In the whole section 5, for each p ∈ R, let Gp be a connected semisimple adjoint Qp-

group without any Qp-anisotropic factors and Dp a maximal compact open subgroup

for almost all p ∈ Rf . We will say that (GA, Dp) has a Q-form (resp. Q-isotropic form)

if there exists a connected semisimple adjoint (resp. Q-isotropic) Q-group H and a

Qp-isomorphism fp : H → Gp for each p ∈ R such that fp(Dp) = H(Zp) for almost all

p ∈ Rf . If (GA, Dp) has a Q-form, we denote by GA(Q) (resp. GA(Q)+) the image of

H(Q) (resp. H(Q)+) under the restriction of∏

p∈R fp to (GA, Dp).

22 HEE OH

5.1. Theorem. Assume that rankRG∞ ≥ 2. Let P1p, P2p be a pair of proper opposite

parabolic Qp-subgroups of Gp for each p ∈ R. Let Γ be a subgroup of (GA, Dp) containing

lattices in Ru(P1)A and Ru(P2)A respectively, where Ru(Pi)A = (GA, Dp)∩∏

p∈R Ru(Pip)

for each i = 1, 2. Assume that (∗) Γ∞ is a lattice in G∞(R). If Γ is discrete, then

(GA, Dp) has a Q-isotropic form such that GA(Q)+ ⊂ Γ ⊂ GA(Q).

Proof. Set Fi = Γ ∩ Ru(Pi)A. By the assumption, Fi is a lattice in Ru(Pi)A. For

any finite subset S of R containing ∞, ΓS is a discrete subgroup of GS , and FSi is

a lattice in Ru(Pi)A ∩ GS for each i = 1, 2 by Lemma 3.1. Under the hypothesis, it

follows from [Oh1] that the subgroup ΓS is a non-uniform S-arithmetic lattice in GS .

Applying Theorem 4.5, we obtain a Q-form on GA such that Γ ⊂ GA(Q). Without loss

of generality, we may assume that there exists a connected absolutely simple Q-group

H such that Γ ⊂ H(Q) and both P1p and P2p are parabolic subgroups of H defined over

Qp for each p ∈ R. Since prp(γ) = prq(γ) for any γ ∈ Γ and for any p, q ∈ R, we have

prp(Γ ∩ (Ru(Pi))A) ⊂ Ru(Pip) ∩ Ru(Piq).

On the other hand, since Γ∩ (Ru(Pi))A is a lattice in (Ru(Pi))A, it follows that prp(Γ∩

(Ru(Pi))A) is Zariski dense in Ru(Pip) for each p ∈ R (c.f. [Lemma 2.3, Oh1]). Hence

Ru(Pip) = Ru(Piq) for any p, q ∈ R. For some fixed prime p ∈ R, set U1 = Ru(P1p)

and U2 = Ru(P2p). Then prp(Γ∩Ui) is Zariski dense in Ui, and hence Ui is defined over

Q. It follows that the Q-form on H is isotropic. Since Γ ∩ (Ui)A ⊂ Ui(Q) and both are

lattices in (Ui)A, Γ∩ (Ui)A has a finite index in Ui(Q). But Ui(Q) is a unipotent group,

and hence it has no finite index subgroup except itself. Therefore Γ∩ (Ui)A = Ui(Q). It

is then well known that the subgroup of H(Q) generated by U1(Q) and U2(Q) coincides

with H(Q)+ (see [BT1]). Therefore Γ contains H(Q)+. This finishes the proof.

5.2. Corollary. Let P1, P2 and G∞ be same as in the above theorem. Let Fi be a

lattice in Ru(Pi)A for each i = 1, 2. Denote by ΓF1,F2the subgroup of (GA, Dp) generated

by F1 and F2. Assume that (∗) Γ∞F1,F2

is a lattice in G∞(R). Then ΓF1,F2is discrete

if and only if there exists a Q-form on GA such that Fi = GA(Q) ∩ Ru(Pi)A for each

i = 1, 2 and ΓF1,F2= GA(Q)+.

Remark 5.3. Theorem 5.1 and Corollary 5.2 hold without the assumption (∗) for any

group G∞ for which Margulis’s conjecture (see [Oh1, Conjecture 0.1]) has been verified.

Indeed, for the subgroup Γ in Theorem 5.1 (or for ΓF1,F2in Corollary 5.2), Γ∞ is a

discrete subgroup containing lattices in a pair of opposite horospherical subgroups in

G∞(R). See the remark following Theorem 1.5.

Hence we obtain the following:


5.4. Corollary. Let G∞ be R-split, rank G∞ ≥ 2 and G∞(R) not locally isomorphic

to SL3(R). Then the following sets are all equal:

(1) discrete subgroups in GA containing lattices in opposite horospherical subgroups

of GA;

(2) subgroups generated by all unipotent elements of a non-uniform irreducible lattice

in GA;

(3) subgroups of f(H(Q)) containing f(H(Q))+ for some H and f as in Theorem

1.1.

The above sets are non-empty only when the adele group (GA, Dp) is isomorphic to

(HA, H(Zp)) as a topological group for some connected absolutely simple Q-isotropic

Q-group H.

6. Lattices in (GA, G(Zp)).

6.1. Let G be a connected absolutely almost simple Q-group. Recall that for any field

k, the non-isomorphic k-forms of a k-algebraic variety M are parametrized by the first

Galois cohomology set H1(k, Aut(M)) (cf. [PR, 2.2.3]). Therefore the number of non-

isomorphic Q-forms of G is determined by the following question on the Hasse principle

for Aut(G): what is the size of the kernel (the Shafarevich-Tate group of G) of the

natural map

(*) H1(Q, Aut(G)) →∏

p∈R

H1(Qp, Aut(G))?

As remarked in [Se, 4.6], a theorem of Borel [Bo, Theorem 6.8] implies that the above

kernel is always finite.

If G does not have any outer automorphism, for instance, if G is not of type An (n ≥

2), Dn (n ≥ 4) or E6, then Aut(G) = Int(G), which is canonically isomorphic to Gad.

Then the Hasse principle for an adjoint Q-group (see [PR, Theorem 6.22]) says that the

kernel of the above map is trivial. We say that a connected absolutely almost simple

Q-group H is a Q-form of GA if for each p ∈ R, H and G are isomorphic over Qp. Hence

we summarize:

Proposition. Let G be a connected absolutely almost simple Q-group. Then GA admits

only finitely many non-isomorphic Q-forms. Moreover if G is not of type An (n ≥

2), Dn (n ≥ 4) or E6, then there is a unique Q-form on GA up to Q-isomorphism.

We remark that there exists two central simple division algebras over Q of degree at

least 2, say, D1 and D2 such that PSL1(D1) and PSL1(D2) are not isomorphic over Q,

24 HEE OH

but isomorphic over all Qp, p ∈ R. Hence the adele group associated with PSL1(D1)

has (at least) two non-isomorphic Q-forms.

6.2. If G is a connected absolutely simple Q-isotropic Q-group, G has no Qp-anisotropic

factors for each p ∈ R. Hence by Lemma 4.3, we have:

Proposition. Let G be a connected absolutely simple Q-isotropic Q-group. Let f be

a topological group automorphism of (GA, G(Zp)). Then there exist Qp-automorphisms

fp of Gp’s, p ∈ R with fp(G(Zp)) = G(Zp) for almost all p ∈ Rf such that f is the

restriction of∏

p∈R fp.

By Theorem 1.1, the above proposition yields the following:

6.3. By Theorem 1.1 and Proposition 6.1, we have:

Proposition. Let G be a connected absolutely simple Q-isotropic Q-group. Then up to

automorphism of (GA, G(Zp)), the adele group (GA, G(Zp)) admits only finitely many

non-uniform irreducible lattices up to commensurability. Moreover if G is not of type

An (n ≥ 2), Dn (n ≥ 4) or E6. then (GA, G(Zp)) admits a unique non-uniform irre-

ducible lattice up to commensurability and up to automorphism of (GA, G(Zp)).

6.4. Recall that for a linear algebraic R-group G, (H, f : H → G) is called an R/Q-form

of G or simply Q-form of G if H is a linear algebraic Q-group and f is an isomorphism

defined over R. For any connected semisimple R-group G with G(R) non-compact, G(R)

admits a Q-form with rank Q(G) = 0; hence a uniform (arithmetic) lattice G(Z), as con-

structed by Borel [Bo1]. It also admits a Q-form with the same Q-rank as rankR(G);

hence a non-uniform arithmetic lattice G(Z). This readily follows from [Oh1, Proposi-

tion 1.4.2] whose proof is due to Prasad. His proof was not delivered clearly therein.

We take this opportunity to give a short proof provided by him.

Proposition (cf. [Oh1, Proposition 1.4.2]). Let G be connected adjoint semisimple

linear algebraic group defined over R. Then for a given minimal parabolic R-subgroup P ,

there exists a Q-form on G with respect to which every parabolic R-subgroup containing

P is defined over Q.

Proof. It is not difficult to reduce to the case when G is absolutely simple (see [Oh1]).

Let Gq be the adjoint Q-split group if the R-form of G is inner or the quasi-split Q-

form of G, splitting over k = Q(i) otherwise. Let P q be the corresponding parabolic

subgroup to P of Gq. By [Se, Proposition 37], G is then obtained from Gq by twisting

by a P q-valued cocycle, say, c on Gal(C/R). By [PR, Proposition 6.17], the natural map


H1(Gal(Q/Q), P q) → H1(Gal(C/R), P q) is surjective. Hence there exists a P q-valued

cocycle d on Gal(Q/Q) whose restriction to Gal(C/R) is is cohomologous to c. Naturally

we may regard this cocycle d as an element of H1(Gal(Q/Q), Gq). Then the twist of

Gq by the cocycle d coincides with the R-form of G over R and its distinguished orbits

of Gal(Q/Q) contain all the distinguished orbits of Gal(C/R) of the R-form of G (cf.

[Oh1, 1.4.1]). This proves our claim.

If G is a connected non-compact semisimple linear algebraic group over Qp, Tamagawa

showed that G(Qp) does not admit any non-uniform lattice [Ta]. However it always

admits a uniform lattice as shown by Borel and Harder [BH].

References

[Bo1] A. Borel, Some finiteness of properties of adele groups over number fields, Publ. Math. IHES16 (1963), 1–30.

[Bo2] A. Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122.[BH] A. Borel and G. Harder, Existence of discrete cocompact subgroups of reductive groups over

local fields, J. Reine Angew. Math 298 (1978), 53–64.

[BT1] A. Borel and J. Tits, Elements unipotents et sous-groupes paraboliques de groupes reductifs I,Invent. Math 12 (1971), 95–104.

[BT2] A. Borel and J. Tits, Homomorphismes “abstraits” de groupes algebriques simples, Ann. Math

97 (1973), 499–571.[Ma1] G. A. Margulis, Arithmeticity of irreducible lattices in semisimple groups of rank greater than

1, appendix to Russian translation of M. Raghunathan, Discrete subgroups of Lie groups (1977(in Russian)), Mr, Moscow; Inven. math 76 (1984), 93-120.

[Ma2] G. A. Margulis, Discrete subgroups of semisimple Lie groups, Berlin Heidelberg New York,

Springer-Verlag, 1991.[MT] G. D. Mostow and T. Tamagawa, On the compactness of the arithmetically defined homogeneous

spaces, Ann. Math 76 (1962), 440–463.

[Oh1] H. Oh, S-arithmeticity of discrete subgroups containing lattices in horospherical subgroups, DukeMath J. 101 (2000), 333–340.

[Oh2] H. Oh, Discrete subgroups generated by lattices in opposite horospherical subgroups, J. Algebra

203, (1998), 621–676.[PR] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, 1994.

[Se] J-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997.[Ta] T. Tamagawa, On discrete subgroups of p-adic algebaric groups, Arithmetical algebraic geometry

(Proc. Conf. Purdue Univ, 1963) (1965), Harper & Row, New York, 11–17.

[Ti] J. Tits, Reductive groups over local fields, Automorphic Forms, Representations, and L-functions33 (1977), Proc. Symp. Pure. Math. AMS, 29–69.

[Zi] R. Zimmer, Ergodic Theory and Semisimple Groups, Boston Basel Stuttgart, Birkhauser, 1984.

Mathematics Department, Princeton University, Princeton, NJ 08544, USA

E-mail address: [email protected]

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