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 including infinite one ∞ and set Q ∞ = R. Let S be any subset of R. For each p ∈ S, let G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors and D p ⊂ G p (Q p ) be a compact open subgroup for almost all finite prime p ∈ S. Let (G S ,D p ) denote the restricted topological product of G p (Q p )’s, p ∈ S with respect to D p ’s. Note that if S is finite, (G S ,D p )= Q p∈S G p (Q p ). We show that if P p∈S rank Q p (G p ) ≥ 2, any irreducible lattice in (G S ,D p ) is a rational lattice. We also present a criterion on the collections G p and D p for (G S ,D p ) to admit an irreducible lattice. In addition, we describe discrete subgroups of (G A ,D p ) 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 R f the set of finite prime numbers, i.e., R f = R −{∞}. We set Q ∞ = R. For each p ∈ R, let G p be a non-trivial connected semisimple algebraic Q p -group and for each p ∈ R f , let D p be a compact open subgroup of G p (Q p ). The adele group of G p ,p ∈ R with respect to D p ,p ∈ R f is defined to be the restricted topological product of the groups G p (Q p ) with respect to the distinguished subgroups D p . We denote this group by (G A , {D p ,p ∈ R f }) or simply by (G A ,D p ). That is, (G A ,D p )= {(g p ) ∈ p∈R G p (Q p ) | g p ∈ D p for almost all p ∈ R f }. As is well known, the adele group (G A ,D p ) is a locally compact topological group. If G is a connected semisimple Q-group, then we mean by (G A ,G(Z p )) the adele group attached to the groups G p = G, p ∈ R with respect to the subgroups G(Z p ), p ∈ R f . It is a well known result of Borel [Bo1] that the diagonal embedding of G(Q) into (G A ,G(Z p )), which we will identify with G(Q), is a lattice in (G A ,G(Z p )). Furthermore 2000 Mathematics Subject Classification number: 20G35, 22E40, 22E46, 22E50, 22E55 1
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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
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).
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 15
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]:
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 17
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
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 19
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
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 21
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:
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 23
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
ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 25
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].
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local fields, J. Reine Angew. Math 298 (1978), 53–64.
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Mathematics Department, Princeton University, Princeton, NJ 08544, USA