ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM. Hee Oh

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1 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

2 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 (G A, G(Z p )). In the spirit of Margulis arithmeticity theorem [Ma1], we show in this paper that any irreducible lattice in an adele group (G A, D p ) is essentially of the form described as above. We say that a lattice Γ in (G A, D p ) is irreducible if, for any finite subset S of R containing, π S (Γ {(g p ) G A g p D p for all p / S}) is an irreducible lattice in p S G p(q p ) in the usual sense (see [Ma2, Ch III, 5.9] or definition 2.9 below) where π S denotes the natural projection (G A, D p ) p S G p(q p ). The following is a sample case of our main theorem: 1.1. Theorem. For each p R, let G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors and D p a compact open subgroup for almost all p R f. Assume that G is absolutely simple. Then any irreducible non-uniform lattice Γ in (G A, D p ) is rational in the sense that there exist a connected absolutely simple Q-isotropic Q-group H and a Q p -isomorphism f p : H G p for each p R with f p (H(Z p )) = D p for almost all p R f such that Γ is a subgroup of finite index in f(h(q)) where f is the restriction of the product map p R f p to (H A, H(Z p )). In particular, f provides a topological group isomorphism of (H A, H(Z p )) to (G A, D p ). In order to define a rational lattice in an adele group in generality, we first describe arithmetic methods of constructing irreducible lattices in adele groups. Let K be a number field. Let R K be the set of all (inequivalent) valuations of K. For each v R K, K v denotes the local field which is the completion of K with respect to v and for non-archimedean v R K, O v denotes the ring of integers of K v. If H is a connected absolutely simple K-group, it is a well known fact that the set T (H) = {v R K H(K v ) is compact} is finite. Let S be a subset of R K T (H) containing all archimedean valuations in R K T (H), and let (H S, H(O v )) denote the restricted topological product of the groups H(K v ), v S with respect to the subgroups H(O v ). Then the subgroup H(K(S)), when identified with its image under the diagonal embedding into (H S, H(O v )), is a lattice in (H S, H(O v )) 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 (H S, H(O v )). Unless mentioned otherwise, throughout the introduction, we let G p be a connected semisimple adjoint Q p -group for each p R and D p a compact open subgroup for each p R f.

3 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Definition. We call an irreducible lattice Γ in (G A, D p ) rational if there exist K, H, S as above and a topological group epimorphism f : (G A, D p ) (H S, H(O v )) with compact kernel such that f(γ) is commensurable with H(K(S)). Remark. (1) Since S R K T (H), H(K v ) is non-compact for each v S. If we denote by G i p the maximal connected normal Q p -subgroup of G p without any Q p -anisotropic factors for each p R and let D i p = D p G i p for each p R f, then in the above definition the quotient (G A, D p )/kerf is isomorphic to (G i A, Di p). In particular, if G p (Q p ) has no compact factors for any p R, we may assume that f is an isomorphism in Definition 1.2. (2) If R 0 = {p R G p (Q p ) is non-compact}, then (G i A, Di p ) is naturally identified with the restricted topological product of the groups G i p (Q p), p R 0 with respect to the subgroups D i p. If R 0 is finite, then (G i A, Di p ) = p R 0 G i p (Q p). In this case, the above definition of a rational lattice in (G A, D p ) coincides with that of an R 0 -arithmetic (usually referred to as S-arithmetic ) lattice of p R 0 G i p(q p ) given in [Ma2, Ch IX, 1.4]. (3) If Γ is an irreducible lattice in (G A, D p ), then pr(γ) is an irreducible lattice in (G i A, Di p ) as well where pr denotes the natural projection (G A, D p ) (G i A, Di p ). Then an irreducible lattice Γ in (G A, D p ) is rational if and only if pr(γ) is a rational lattice in (G i A, 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 Main Theorem. If p R rank Q p (G p ) 2, any irreducible lattice in (G A, D p ) is rational. That the adele group (G A, D p ) contains an irreducible lattice imposes a strong restriction not only on the family of the ambient groups G p but also on the family of distinguished subgroups D p. The following presents a necessary and sufficient condition on those restriction: 1.4. Theorem. For each p R, assume that G p (Q p ) has no compact factors. The adele group (G A, D p ) admits an irreducible lattice if and only if there exist a connected semisimple Q-simple Q-group H such that G p is Q p -isomorphic to a connected normal Q p -subgroup of H for each p R and D p is a subgroup whose volume is maximum among all compact open subgroups of G p (Q p ) for almost all p R f. See Theorem 4.13 below for a more general statement.

4 4 HEE OH Example. (1) If G p is Q p -simple and Q p -isotropic for each p R and (G A, D p ) admits an irreducible lattice, then all G p s are typewise homogeneous, that is, their Dynkin types are the same. (2) Let n 2 and G p = PGL n for each p R. Then (G A, D p ) has an irreducible lattice if and only if D p is conjugate to PGL n (Z p ) for almost all p R f. For n = 2, for each p R f, there are two conjugacy classes of maximal compact open subgroups of PGL 2 (Q p ), represented by PGL 2 (Z p ) and by {( ) } ( ) a b 0 1 L p = PGL pc d 2 (Z p ), p 0 respectively. Note that in the Bruhat-Tits tree associated to PGL 2 (Q p ), the conjugacy class of PGL 2 (Z p ) corresponds to the stabilizer of a vertex and the conjugacy class of L p corresponds to the stabilizer of the middle point of an edge. If we denote by µ p a Haar measure of PGL 2 (Q p ), then µ p (L p ) = 2 p+1 µ p(pgl 2 (Z p )) {( ) } a b because the common subgroup PGL pc d 2 (Z p ) has index p + 1 in PGL 2 (Z p ) while it has index 2 in L p. Hence if D p is conjugate to L p for infinitely many primes p, (G A, D p ) does not admit an irreducible lattice. Furthermore, it follows from Theorem 1.1 and the Hasse principle that, up to automorphism of (G A, PGL 2 (Z p )), PGL 2 (Q) is the unique irreducible non-uniform lattice in (G A, PGL 2 (Z p )) up to commensurability (see Proposition 6.3 below). (3) Let n 1. If G p = PGSp 2n for each p R, then (G A, D p ) admits an irreducible lattice only when D p is conjugate to PGSp 2n (Z p ) for almost all p R f. Up to automorphism of (G A, PGSp 2n (Z p )), the subgroup PGSp 2n (Q) is the unique irreducible non-uniform lattice in (G A, PGSp 2n (Z p )) 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 R f, G(Z p ) is a hyperspecial subgroup of G(Q p ), or equivalently, the volume of G(Z p ) is the maximum among all compact open subgroups of G(Q p ) [Ti]. It thus follows from Theorem 1.4 that if we let G p = G, p R, then (G A, D p ) admits an irreducible lattice if and only if D p is conjugate to G(Z p ) for almost all p R f (see 4.14). If H is a connected semisimple Q-isotropic Q-group, there exists a pair P 1, P 2 of opposite proper Q-parabolic subgroups of H. Then the subgroup R u (P i )(Q) is a (uniform) lattice in (H A, H(Z p )) p R R u(p i )(Q p ) where R u (P i ) denotes the unipotent radical

5 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 5 of P i 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 R u (P 1 )(Q) and R u (P 2 )(Q) coincides with the subgroup H(Q) + [BT1]. We also show that a discrete subgroup of (G A, D p ) containing any lattices 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 (G A, D p ) is called a horospherical subgroup if U = (G A, D p ) p R R u(p p ) where P p is a proper parabolic Q p -subgroups of G p for each p R. Two horospherical subgroups of (G A, D p ) are called opposite if the corresponding Q p - parabolic subgroups are opposite for each p R. For a subgroup Γ (G A, D p ), we denote by Γ the image of Γ (G (R) p R f D p ) under the natural projection pr : (G A, D p ) G (R) Theorem. For each p R, assume that G p has no Q p -anisotropic factors. Assume that rank (G ) 2. Let Γ be a subgroup of (G A, D p ) containing lattices in a pair of opposite horospherical subgroups of (G A, D p ). 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 : (H A, H(Z p )) (G A, D p ) 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 SL 3 (R). We also remark that in an S-arithmetic setting (S finite), i.e., when G = p S G p(q p ), 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 (G A, D p ) can admit up to commensurability. By the Hasse principle for an adjoint absolutely simple Q-group, Theorem 1.1 implies that for instance, if for some p R, G p is not of

6 6 HEE OH type A n (n 2), D n (n 3), or E 6, then (G A, D p ) admits at most one irreducible non-uniform lattice up to commensurability and up to automorphism of (G A, D p ) (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 D p so that (G A, D p ) 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 Q p -anisotropic factors, H is isomorphic to G p, say via f p, for each p R and H(Q) corresponds to Γ under the product map p R f p, up to commensurability. Here we embed Γ into p R f p(h(q)) (G A, D p ) 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 D p s. To ensure the Q p -isomorphisms f p s transfer the Q-structure on H to (G A, D p ) in a compatible way, we has to show that f p (H(Z p )) = D p for almost all p R f. 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 (G A, D p ). Acknowledgment. Thanks are due to Gopal Prasad and Andrei Rapinchuk for communications 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 For any field k, we mean by a linear algebraic k-group M that M is a Zariski closed k-subgroup of GL N for some N, and we set M(J) = M GL N (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 GL N for some N Let K be a number field. Let R K, K v and O v be as in the introduction. For any S R K, we set K(S) to be the ring of S-integers in K, that is, K(S) = {x K x O v for all v R f S}. If K = Q, we simply set R = R Q and we sometimes write Z S

7 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 7 for Q(S) For each v R K, let G v be a connected semisimple algebraic K v -group. Fix a compact open subgroup D v of G v (K v ) for each non-archimedean v R K. For any subset T R K, we denote by (G T, D v ) the restricted topological product of the groups G v (K v ), v T with respect to the distinguished subgroups D v. If T is finite, then (G T, D v ) = v T G v(k v ), which we will then simply denote by G T. If T = R K, we write (G A, D v ) = (G T, D v ) and call an adele group. The topology on (G T, D v ) is given as follows: a base of open subsets consists of the sets of the form p S U v v/ S D v where S is a finite subset of T and U v G v (K v ) is an open subset for each v S. The group (G T, D v ) is a locally compact group with respect to this topology As is well known, for each v R K, the local field K v is a finite extension of a subfield isomorphic to Q p for some (unique) p R. For each p R, we denote by I p the set of valuations v R K such that Q p K v (up to isomorphism). Then H p = v I p Rest Kv /Q p G v is a connected semisimple Q p -group and v I p G v (K v ) is isomorphic to H p (Q p ) as topological groups. We denote this isomorphism by Rest 0. The map Rest 0 also extends to an isomorphism of the adele group (G A, D v ) with the adele group (H A, M p ) where M p = v I p Rest Kv /Q p D v (cf. [Ma2, Ch I, 3.1.4] and [Bo1]) If G is a connected semisimple K-group, then we mean by (G A, G(O v )) the adele group attached to the groups G v = G with respect to the subgroups G(O v ). The diagonal embedding of G(K) into (G A, G(O v )), which we will identify with G(K), is a lattice in (G A, G(O v )) [Bo1]. Denote by T (G) the set of all v R K such that G(K v ) is compact. Then T (G) is finite (cf. [Ma2, Ch I, 3.2.3]). It then follows that if T R K contains all archimedean valuations in R K T (G), the subgroup G(K(T)) is a lattice in (G T, G(O v )) when diagonally embedded into (G T, G(O v )). Note that G(K(T)) = {x G(K) x G(O v ) for all non-archimedean v R K T }. The group G is K-isotropic if and only if G(K(T)) is a non-uniform lattice in (G T, G(O v )) For each p R, we denote by pr p the natural projection (G A, D p ) G p (Q p ). For any T R, the notation pr T denotes the natural projection (G A, D p ) (G T, D p ). We set G A(T) = {(g p ) G A g p D p for all p R f T }. For any subgroup H of (G A, D p ) and for any subset T R, we set H T = pr T ( H GA(T) ).

8 8 HEE OH 2.7. For any finite S R, let G p be a connected semisimple algebraic Q p -group without any Q p -anisotropic factors for each p S. A lattice Λ in G S = p S G p(q p ) is called irreducible if for any two connected normal subgroups H = p S H p(q p ) and M = p S M p(q p ) of G S such that for each p S, G p is an almost direct product of connected normal Q p -subgroups M p and H p, (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 For a connected semisimple Q p -subgroup G p, we denote by G i p (i standing for isotropic) the maximal connected normal subgroup of G p without any Q p -anisotropic factors. Note that G p (Q p )/G i p (Q p) is compact. That G p has no Q p -anisotropic factors is equivalent to saying that G p (Q p ) has no compact factors Let G p be a connected semisimple algebraic Q p -group for each p R. Fix a compact open subgroup D p of G p (Q p ) for each p R f. Let T R. A lattice Γ in (G T, D p ) is called irreducible if the projection of Γ S into p S G p(q p ) i is an irreducible lattice in p S G p(q p ) i for any finite subset S T containing For a connected semisimple algebraic K-group G, the notation G(K) + denotes the normal subgroup of G(K) generated by the subgroups R u (P)(K) where P runs through the set of all parabolic K-subgroups of G and R u (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 generated by R u (P 1 )(K) and R u (P 2 )(K) for any pair P 1, P 2 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 We refer to [PR], [Bo1] or [Ma2, Ch I] as a general reference to our terminology regarding algebraic groups. 3. Distinguished subgroups D p 3.1. Lemma. Let G p be a connected semisimple Q p -group for each p R. Fix a compact open subgroup D p of G p (Q p ) for each p R f. 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 (G A, D p ), then Γ S is a discrete subgroup in G S. (2) If Γ is a (uniform) lattice in (G A, D p ), then Γ S is a (uniform) lattice in G S.

9 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 9 Proof. For simplicity, set G A = (G A, D p ). Denote by pr S the restriction of pr S to the subgroup G A(S). Since the kernel of pr S is compact, the subgroup Γ S is discrete if Γ is so. Let Γ be a (uniform) lattice in G A. Since G A(S) is an open subgroup of G A, the intersection Γ G A(S) is a (uniform) lattice in G A(S). Consider the natural map pr S : G A(S) /(Γ G A(S) ) G S /Γ S induced by pr S. Now if µ is an invariant measure on G A(S) /(Γ G A(S) ), then pr S (µ) is an invariant measure on G S/Γ S. Hence if Γ is a uniform lattice in G A, then Γ S is a uniform lattice in G S 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 G p(q p ) is non-compact. Then G(Z S ) is dense in the direct product p R f S G(Z p), when diagonally embedded. In particular if is a subgroup of finite index in G(Z S ), then the closure of has finite index in p R f S G(Z p). Proof. Let G AS denote the restricted topological product of the G p (Q p ) for p R f S with respect to the subgroups G(Z p ), p R f S. By the strong approximation theorem (cf. [PR, Theorem 7.12, P. 427]), G(Q) is dense in G AS. Since p R f S G(Z p) is an open subgroup of G AS, G(Q) p R f S G(Z p) is dense in p R f S G(Z p). Since G(Z S ) = G(Q) p R f S G(Z p), this proves the first claim. For the second claim, it suffices to note that [G(Z S ) : ] [G(Z S ) : ] by the first claim Proposition. Let p R f. Let G be a connected semisimple Q p -group and let K 1 and K 2 be maximal compact subgroups of G(Q p ). Assume that there exists a maximal compact subgroup K of G(Q p ) such that π(k) K 1 K 2 where G is the simply connected covering of G and π : G G is the Q p -isogeny. Then K 1 = K 2. 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(Q p ) acts on B through the map π. For each i = 1, 2, the maximal compact subgroup K i is the stabilizer G(Q p ) x i in G(Q p ) of some point x i B. Since G is simply connected, there exists a vertex v B such that G(Q p ) v = K. We claim that v = x 1 = x 2, which implies that K i G(Q p ) v for both i = 1 and 2. Since K 1 and K 2 are maximal compact subgroups, this implies that K 1 = K 2 = G(Q p ) v. Suppose that v x i for some i {1, 2}. Since π(k) stabilizes v and x i, it stabilizes pointwisely the unique geodesic l joining v and x i. Since v x i, we can find a facet F whose closure contains v and F l is non-empty. Fix z F l. Note

10 10 HEE OH that the dimension of F is positive. Since G is simply connected, G(Qp ) z = G(Q p ) F where G(Q p ) F is the pointwise stabilizer of F in G(Q p ). Therefore π(k) G(Q p ) F. This is a contradiction, since the stabilizer of a facet of positive dimension in G(Q p ) cannot be a maximal compact subgroup. This finishes the proof Lemma [Ti, 3.2]. Let G be a connected semisimple Q-group. Then G(Z p ) is a maximal compact subgroup for almost all p R f 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(Z p )) G(Z p ). See [PR, P. 451] for the proof of Lemma Proposition. Let G, G and π be as in Lemma 3.5. For each p R f, let D p be a compact open subgroup of G(Q p ). Assume that π( G(Z p )) D p for almost all p R f. Then D p G(Z p ) for almost all p R f. Proof. Since every compact open subgroup of G(Q p ) is contained in a maximal compact open subgroup, we may assume that D p is a maximal compact open subgroup for all p R f. By Lemmas 3.4 and 3.5, there exists a finite subset S R f such that for each p R f S, both D p and G(Z p ) are maximal compact subgroups of G(Q p ) and π( G(Z p )) D p G(Z p ). Applying Proposition 3.3, we have D p = G(Z p ) for all p R f S, proving the claim 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 Q p G 2. If G(R) is non-compact, we assume that S. If Γ G(Q) and Γ is a lattice in G S (when diagonally embedded), then Γ and G(Z S ) are commensurable. In particular Γ contains a subgroup of finite index in G(Z S ) Set G(Q) (G A, D p ) = {x G(Q) x D p for almost all p R f }. We identify this set with its image under the diagonal embedding into (G A, D p ). Theorem. Let G and S be as in Lemma 3.7. Let D p be a compact open subgroup of G(Q p ) for each p R f. If Γ is a closed subgroup of G(Q) (G A, D p ) such that Γ S is a lattice in G S = p S G(Q p), then π( G(Z p )) D p G(Z p ) for almost all p R f, where G and π are as in Lemma 3.5. Furthermore if Γ is a lattice in (G A, D p ), then D p = G(Z p ) for almost all p R f.

11 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 11 Proof. We first consider the case when G is simply connected. By Theorem 3.7, Γ S contains a subgroup of finite index in G(Z S ). Let denote the diagonal embedding of the subgroup Γ S G(Z S ) into p R f S G(Z p). Note that p R f S (G(Z p) D p ) p R f S G(Z p). By Theorem 3.2, the closure of is a compact open subgroup in p R f S G(Z p), and hence G(Z p ) pr p ( ) for almost all p R f. On the other hand, since p R f S (G(Z p) D p ) is compact, we have pr p ( ) G(Z p ) D p for each p R f S. Therefore G(Z p ) = D p for almost all p R f 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 π p the restriction of π to G(Q p ). By Lemma 3.5, we have that G(Z p ) = π 1 p G(Z p) for almost all p R f. Set Γ = G(Q) ( G A, π 1 p (D p)). Since the kernel of p S π p is finite, it is clear that Γ S is a lattice in p S G(Q p ). Therefore, by the previous simply connected case, we have πp 1(D p) = G(Z p ); hence π p ( G(Z p )) D p for almost all p R f. Therefore for almost p R f, π p ( G(Z p )) D p G(Z p ) by Proposition 3.6. Now assume that Γ is a lattice in (G A, D p ). Recall that G(Q) is a lattice in (G A, G(Z p )). Denote by µ 1 and µ 2 the Haar measures on (G A, D p ) and (G A, G(Z p )) normalized so that µ 1 (D p ) = 1 and µ(g(z p )) = 1 for all p R f, respectively. For each finite S R containing, Γ S is a subgroup of finite index in G(Z S ) and both are lattices in G S. Hence µ 2 (G S /Γ S ) µ 2 (G S /G(Z S )). Note that µ 2 (G S /Γ S ) = µ 1 (G S /Γ S ) p R f S [G(Z p) : D p ]. For any increasing sequence S i, i = 1, 2, such that R = i S i, the measures µ 2 (G Si /G(Z Si )) and µ 1 (G Si /Γ S i ) converge to the measures µ 2 ((G A, G(Z p ))/G(Q)) and µ 1 ((G A, D p )/Γ) respectively. Hence lim i p R f S i [G(Z p ) : D p ] should be bounded. Hence D p = G(Z p ) for almost all p R f Theorem. Let G be a connected semisimple almost Q-simple group and D p a compact open subgroup of G(Q p ) for each p R f. If G is not simply connected, assume that D p is a maximal compact open subgroup for almost all p R f. Then the following are equivalent. (1) The adele group (G A, D p ) admits a lattice contained in G(Q) (G A, D p ). (2) There exists a finite subset S R such that p S rank Q p G 2, and Γ S is a lattice in p S G(Q p) where Γ = G(Q) (G A, D p ). (3) D p = G(Z p ) for almost all p R f.

12 12 HEE OH Proof. To show (1) (2), since G(Q p ) is non-compact for almost all p R (see 2.5), there exists a finite subset S R such that p S rank Q p G 2. If G (R) is noncompact, we assume S. Let be a lattice in (G A, D p ) contained in G(Q). It is not difficult to check that Γ is a discrete subgroup of (G A, D p ). Since Γ, the subgroup Γ is a lattice in (G A, D p ) as well. Hence Γ S is a lattice in p S G(Q p) by Lemma 3.1. The direction (2) (3) follows from Theorem 3.8. If (3) holds, then (G A, D p ) = (G A, G(Z p )) in the sense that the identity map provides a topological group isomorphism between them, and hence G(Q) is a lattice in (G A, D p ) 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 D p a compact open subgroup of G(Q p ) for each p R f. Then the following are equivalent. (1) The adele group (G A, D p ) admits a lattice contained in G(Q) (G A, D p ). (2) D p = G(Z p ) for almost all p R f. 4. Rationality of an irreducible lattice in G A 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 G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors for each p T and D p a compact open subgroup of G(Q p ) for almost all finite p T. Definition A. An irreducible lattice Γ in (G T, D p ) 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 = Hp 1 H2 p where H1 p and H2 p are connected semisimple adjoint Q p -groups; (4) for each p T, a maximal compact open subgroup M p Hp(Q 2 p ) with M p = H(Z p ) Hp 2(Q p) for almost all finite p T; and (5) a family of Q p -epimorphisms f p : H G p, p T with kerf p = Hp 2 and f p (H(Z p )) = D p for almost all finite p T such that Γ is commensurable with the subgroup f H(Q(T)) p T(H p 1 (Q p) M p )

13 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 13 where f = p T f p. Definition B. An irreducible lattice in (G T, D p ) is called a rational lattice if there exist a number field K, a connected absolutely simple K-group H and a subset B R K T (H) containing all archimedean valuations in R K T (H) such that there exists a topological group isomorphism f : (H B, H(O v )) (G T, D p ) 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 ] 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 (G T, D p ) 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 = Rest K/Q H. For each p T, let I p R K be as in 2.4. Then considering H as a Q p -group and H as a K v -group for each v I p, we have the decomposition H = v I p Rest Kv /Q p H over Q p. Since H is a connected absolutely simple K v -group, each Rest Kv /Q p H is a connected semisimple adjoint Q p -simple Q p -group. For each p T, we can find a partition of I p into I 1 p I2 p such that H1 p = v I 1 p Rest Kv /Q p H and H 2 p = v I 2 p Rest Kv /Q p H = kerf p. Set B = p T I 1 p. Note that if T, H 2 (R) is compact, since otherwise H 2 (R) does not admit a compact open subgroup. Hence if v R K is an archimedean valuation with H(K v ) non-compact, then v / I. 2 That is, I 1 and hence B contains all archimedean valuations in R K T (H). If / T, H(R) is compact, and hence for all archimedean v I, H(K v ) is compact, that is, v T (H). Since G p (Q p ) has no compact factors, B R K T (H). Since M p = H(Z p ) H 2 p(q p ) for almost all finite p T and M has finite index in H 2 (R) in the case when T, the subgroup Rest K/Q H(K(B)) is commensurable with {x H(Q(T)) x H 1 p (Q p) M p for each p T }. Via the map Rest 0 (see 2.4), the group (H B, H(O v )) is isomorphic to (H 1 T, H(Z p) H 1 (Q p )) and the lattice H(K(B)) is mapped to a subgroup commensurable with the subgroup pr 1 δ T ( {x H(Q(T)) x H 1 p (Q p ) M p for each p T } ) where δ T denotes the diagonal embedding of H(Q) into (H T, H(Z p )) and pr 1 denotes the canonical projection (H T, H(Z p )) (H 1 T, H(Z p) H 1 p (Q p)).

14 14 HEE OH Note that Rest 0 ( v I H(O p 1 v )) = H(Z p ) Hp 1(Q p) for each finite p T. Hence if f 1 denotes the restriction of the map f to (HT 1, H(Z p) H 1 (Q p )), then f 1 Rest 0 is an isomorphism of (H B, H(O v )) to (G T, D p ) 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 = Rest K/Q H, then H is a connected semisimple adjoint Q-simple Q-group. Let T = {p R for some v B, K v is a finite extension of Q p }. Note that H(R) is non-compact if and only if H(K v ) is non-compact for an archimedean valuation v R K. Hence if H(R) is non-compact, then T, since B contains all archimedean valuations in R K T (H). Let Ip 1 = I p B and Ip 2 = I p Ip 1. Set Hp 1 = v I Rest p 1 Kv /Q p H and Hp 2 = v I Rest p 2 Kv /Q p H. Note that if T, H 2 (R) is compact. It follows from the lemma below that there exist Q p-isomorphisms h p : Hp 1 G p, p T such that h p (Hp 1(Q p) H p (Z p )) = D p for almost all finite p T and f = p T h p Rest 0 where f : (H B, H(O v )) (G T, D p ) is the given topological group isomorphism and Rest 0 : v I H(K p 1 v ) Hp 1(Q p) as in 2.4. If pr p denotes the natural projection H Hp, 1 then the map f p = h p pr p is a Q p -epimorphism from H G p with kerf p = Hp 2 and f p(h(z p )) = D p for almost all finite p T. Set M p = H(Z p ) Hp(Q 2 p ) for each finite p T. If T, set M = H (R). 2 Then H(K(B)), is commensurable to the subgroup {x H(K(B 0 )) Rest Kv /Q p x Hp 1 M p for each p T } v I p where B 0 = p T I p. Therefore via the map p T f p, Γ is commensurable to {x H(Q(T)) x Hp 1 (Q p) M p for each p T }. Hence Γ is rational as in Definition A. We formulate the lemma used in the above proof Lemma. Let S, T R. Let G p, p S (resp. H p, p T) be connected semisimple adjoint Q p -groups without any Q p -anisotropic factors and M p G p (Q p ) (resp. L p H p (Q p )) compact open subgroups for each finite prime p S (resp. p T). Assume that M p and L p are maximal compact subgroups for almost all finite p S. If f : (G S, M p ) (H T, L p ) is a topological group isomorphism, then S = T, there exist Q p - isomorphisms f p : G p H p, p S such that f p (M p ) = L p for almost all finite p S and f(x) = p S f p(x) for any x (G S, M p ).

15 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 15 Proof. For each p S and r T, consider the map f pr : G p (Q p ) H r (Q r ) defined by f pr (g) = pr r (f(g)) for each g G p (Q p ). Here pr r : (H T, L p ) H r (Q r ) denotes the natural projection map. Then f pr is a continuous homomorphism for each p and r. Since f is an isomorphism, for each p S, f pr (G r (Q r )) {e} for some r T. By [Ma2, Ch I, Proposition 2.6.1], we have f pr (G r (Q r )) {e} if and only if p = r, and when p = r, the topological group isomorphism f pp : G p (Q p ) H p (Q p ) extends to a rational Q p -isomorphism f p : G p H p. It follows that S = T. Since the restriction of f to (G S Rf, M p ) induces an isomorphism f : (G S Rf, M p ) (H T Rf, L p ), the image f ( p S R f M p ) is an open compact subgroup of (H T Rf, L p ). Since f ( p S R f M p ) = p S R f f p (M p ) and f p (M p ) H p (Q p ) for each p S R f, L p f p (M p ) for almost all finite p S. Since M p and L p are maximal compact for almost all finite p S, we have that L p = f p (M p ) for almost all finite p S Margulis s S-arithmeticity theorem states: Theorem. Let S R be a finite subset and let G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors for each p S. If p S rank Q p (G p ) 2, any irreducible lattice in G S is an S-arithmetic lattice in G S. See [Ma2, Ch IX, Theorem 1.11 and the remark 1.3. (iii)] or [Zi, Theorem ] Before we give a proof of rationality theorem which works for uniform and nonuniform 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 G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors. For each p R f, let D p G p (Q p ) be a compact open subgroup. Assume that G is absolutely simple. Fix a finite subset S 0 R containing such that p S 0 rank Qp (G p ) 2. Let Γ be a subgroup of (G A, D p ) such that Γ S is an irreducible non-uniform lattice in G S for any finite S T including S 0. Then there exist a connected absolutely simple Q-isotropic Q-group H and a Q p -isomorphism f p : H G p for each p R with f p (H(Z p )) = D p for almost all p R f such that Γ f(h(q)) where f is the restriction of p R f p to (H A, H(Z p )). Proof. Set Ω = {S R S 0 S, S < }. Step 1. Obtain Q-forms H S for each S Ω. For any S Ω, by Theorem 4.4 and Definition A, there exist a connected absolutely simple Q-group H S, Q p -isomorphisms

16 16 HEE OH r Sp : H S G p, p S such that Γ S is commensurable with the subgroup {(r Sp (x)) x H S (Z S )}. Since the groups G p, p S and H S are adjoint, we may assume that Γ S is a finite index subgroup of {(r Sp (x)) x H S (Z S )}. Step 2. The Q-forms H S are all Q-isomorphic. Set H = H S0 and r = r S0. 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 r S (H S (Q)) r(h(q)) and r r 1 S, (Γ ) H S (Q), the map r r 1 S : H H S is defined over Q [Ma2, Ch I, 0.11]. Since both H and H S are absolutely simple, the map r r 1 S is indeed a Q-isomorphism. Step 3. Define a Q p -isomorphism f p : H G p for each p R. For each p R, we define a map f p : H G p by f p = r Sp (r S ) 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 S 1, S 2 Ω such that p S 1 S 2, r S1 p r 1 S 1 = r S 2 p r 1 S 2. Since Γ,p {(r S1 (x), r S1 p(x)) x H S1 (Q)} {(r S2 (x), r S2 p(x)) x H S2 (Q)}, we have that r S1 p r 1 S 1 (z) = r S 2 p r 1 S 2 (z) for any z pr (Γ {,p} ). Since pr (Γ {,p} ) is a Zariski dense subset in G, r S1 p r 1 S 1 = r S 2 p r 1 S 2 and hence the map f p is well defined for each p R. Since r Sp is a Q p -isomorphism and (r S ) 1 r is a Q-isomorphism, f p is a Q p -isomorphism. Step 4. Show Γ f(h(q)) where f = p R f p. We now claim that Γ f(h(q)) where f = p R f p and f(h(q)) = {(f p (x)) x H(Q)}. It suffices to show that Γ S f S (H(Q)) = {(f p (x)) p S x H(Q)} for each S Ω. But Γ S {(r Sp (x)) p S x H S (Q)}. If x H S (Q), then there exists a unique y H(Q) such that x = r 1 S r(y). Hence r Sp (x) = f p (y) for each p S. Therefore Γ S f S (H(Q)) for any S Ω. Step 5. Show f p (H(Z p )) = D p for almost all p R f. The product map f induces a topological group isomorphism from (H A, fp 1(D p)) to (G A, D p ). Note that f 1 (Γ) H(Q) (H A, fp 1 (D p )). Since f 1 (Γ) S is a lattice in p S H(Q p) for any S Ω, by Theorem 3.10, we have fp 1(D p) = H(Z p ), or equivalently f p (H(Z p )) = D p, for almost all p R f. This finishes the proof 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]:

17 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 = Hp 1 Hp 2 where the subgroups Hp 1 and Hp 2 are connected normal Q p -subgroups of H, Hp 1 has no Q p-anisotropic factors and M p Hp 2(Q p) 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 (Hp 1(Q p) M p ) for each p S} and δ : Λ F(Q) a homomorphism with a Zariski dense image in F. Assume that p S rank Q p (Hp) 1 2. Then there exists a (unique) Q-isomorphism j : H F which extends δ Lemma. Let S and G p be as on Theorem 4.4. If Γ is an irreducible lattice in G S. Then for any p S, the restriction of pr p : G S G p to Γ is injective. Proof. If S = {p}, the statement is trivial. Suppose not. Set N = {γ Γ pr p (γ) = e}. Then N is a normal subgroup of Γ. Since the lattice Γ is irreducible, the image of Γ under pr p 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 G S. Since the groups G p are adjoint, the center of G S is trivial, proving the claim Lemma. For any p R, let G be a connected reductive Q p -group. Then any compact open subgroup of G p (Q p ) is contained in only a finitely number of compact subgroups of G p (Q p ). 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] We are now ready to prove the main theorem: Theorem. Let T R. For each p T, let G p be a connected semisimple adjoint Q p - group without any Q p -anisotropic factors. For almost all finite p T, let D p G p (Q p ) be a maximal compact open subgroup. Fix a finite subset S 0 T (containing if T) such that p S 0 rank Qp (G p ) 2. Let Γ be a subgroup of (G T, D p ) such that Γ S is an irreducible lattice in G S for any finite S T including S 0. Then Γ is contained in some rational lattice in (G T, D p ). Proof. Set p 0 = if T, and otherwise let p 0 be any fixed prime in S 0. Set Ω = {S T S 0 S, S < }. Step 1. Obtain the Q-forms H S, S Ω. For each S Ω, we denote by H S, HSp 1, HSp 2, M Sp, f Sp, pr Sp as in Theorem 4.4 and Definition A in 4.1. Also set r Sp to be the

18 18 HEE OH composition map f Sp pr Sp : H S G p. Since the groups G p, p S and H S are adjoint, we may assume that Γ S is a subgroup of finite index in {(r Sp (x)) p S x H S (Z S ) (HSp 1 M Sp ) for each p S}. Step 2. The Q-forms H S are all Q-isomorphic. Set H = H S0 and r = r S0 p 0. We first claim that the group H S is Q-isomorphic to H for any S Ω. Consider the maps r : H G p0 and r Sp0 : H S G. For simplicity, we set pr p0 = pr 0. Since pr 0 (Γ S ) r Sp0 (H S (Z S )) for any S Ω, the set r 1 Sp 0 (pr 0 (Γ S )) = {x H S (Z S ) r(x) pr Sp0 (Γ S )} is contained in H S (Z S ). Since the map r Sp0 is injective over r 1 Sp 0 (pr 0 (Γ S 0 )) by Lemma 4.7, the composition map r 1 Sp 0 r is well defined on r 1 (pr 0 (Γ S 0 )), which we denote by j Sp0 : r 1 (pr 0 (Γ S 0 )) H S (Q). By Proposition 4.6, the map j Sp0 extends to a Q-rational isomorphism j S : H H S, proving our claim. Step 3. Define Q p -epimorphisms f p : H G p. For each p T, we define a map f p : H G p by f p = r Sp j S 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 S 1, S 2 Ω such that p S 1 S 2, r S1 p j S1 = r S2 p j S2. Note that r is injective over H(Z S0 ) by Lemma 4.7. We let r 1 (Γ S 0 ) = {x H(Z S0 ) r(x) Γ S 0 }. Since H S0 is Q-simple, it follows that r 1 (pr 0 (Γ S 0 )) is Zariski dense in H S0. Hence it suffices to verify this equality for any x r 1 (pr 0 (Γ S 0 )). There exists a unique (see Lemma 4.7) element y Γ S 0 such that r(x) = pr 0 (y), and there exist elements z i H Si (Q), i = 1, 2 such that y = pr S0 (r S1 q(z 1 )) = pr S0 (r S2 q(z 2 )). Again by Lemma 4.7, we have r S1 p(z 1 ) = r S2 p(z 2 ). Then r S1 p j S1 (x) = r S1 p r 1 S 1 p 0 r(x) = r S1 p(z 1 ) which is equal to r S2 p(z 2 ) = r S2 p r 1 S 2 p 0 r(x) = r S2 p j S2 (x). This proves our claim, yielding that f p is well defined for each p T. Since r Sp is a Q p -epimorphism and j S is a Q-isomorphism, f p is a Q p -epimorphism. Step 4. The groups H 1 p, H2 p and M p. Note that kerf p is a connected semisimple adjoint Q p -group, as is any connected normal Q p -subgroup of H. Letting Hp 2 = kerf p for each p T, there exists a connected semisimple adjoint Q p -group Hp 1 such that H = Hp 1 Hp. 2 Note that the restriction f p : Hp 1 G p is a Q p -isomorphism. Since j S is a Q-isomorphism, Hp 1 = j 1 S (H1 Sp ) and H2 p = j 1 S (H2 Sp ) for each S Ω containing p. We claim that there exists a compact open subgroup, say, M p, of Hp(Q 2 p ) such that f 1 p f 1 p (pr p(γ S )) M p for each S Ω containing p. Note that fp 1(pr p(γ S 0 )) (pr p(γ S )) for each S Ω containing p. On the other hand, the latter subgroup is

19 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 19 contained in the compact open subgroup j 1 S (M Sp) HSp 2 (Q p). Since fp 1(pr p(γ S 0 )) S Ω j 1 S (M Sp), S Ω j 1 S (M Sp) is a compact open subgroup of Hp 2(Q p). Hence by Lemma 4.8, p S Ω j 1 S (M Sp) = j 1 S m (M Sm p) for some S m Ω. It suffices to set M p to be a maximal compact open subgroup of Hp 2(Q p) containing j 1 S m (M Sm p). Step 5. Show Γ f(h(q)) where f = p T f p. We now claim that Γ f(h(q) p T (H1 p M p)) where f = p T f p. It suffices to show that Γ S f S (H(Q) p R (H1 p M p )) for each S Ω, where f S = p S f p. For any γ Γ S, we have γ = (r Sp (x)) p S for some x H S (Q) and in the decomposition x = x 1 p x2 p for x1 p H1 Sp and HSp 2, we have x2 p M p. Since x H S (Q), then there exists a unique y H(Q) such that x = j S (y). Hence r Sp (x) = f p (y) and y = j 1 S (x1 p )j 1 S (x2 p ) where j 1 S (x1 p ) H1 p and j 1 S (x2 p ) j 1 S (ker pr Sp) M p for each p S. Therefore Γ S f S (H(Q) p R (H1 p M p )) for any S Ω. Step 6. f p (H(Z p )) = D p for almost all finite p T. For each p T, set D p = {x H 1 p(q p ) f p (x) D p }. For p T, set L p = D p M p and for p R f T, L p = H(Z p ). Consider the adele group (H A, L p ). Now the subgroup {x H(Q) f T (x) Γ} satisfies the property (2) in Theorem 3.8 where f T = p T f p. Hence L p = H(Z p ) for almost all p R f, and hence we have D p M p = H(Z p ) and M p = H(Z p ) H 2 p (Q p) for almost all finite p T. Therefore f p (H(Z p )) = D p for almost all finite p T. Therefore we have constructed (H, f p, M p ) as required in Definition A Corollary. Let T R. For each p T, let G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors. For each finite p T, let D p G p (Q p ) be a compact open subgroup. If p T rank Q p (G p ) 2. then any irreducible lattice in (G T, D p ) is rational. Proof. The condition on maximality of D p 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 L p = H(Z p ) for almost all p R f. Then the rest proceeds exactly the same way Without the assumption of G p being adjoint, we can deduce the following from Theorem 4.5 and Theorem 3.9: Proposition. For each p R, let G p be a connected semisimple Q p -group without any Q p -anisotropic factors and let G be absolutely almost simple. Let D p be a compact open subgroup of G p (Q p ) for each p R f. If Γ is an irreducible non-uniform lattice in (G A, D p ), then there exists a connected absolutely simple Q-group H and a Q p -isogeny f p : G p H for each p R such that π( H(Z p )) f p (D p ) H(Z p ) for almost p R

20 20 HEE OH and p R f p(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 G p = SL n for each p R. Let D p be a (not necessarily maximal) compact open subgroup of SL n (Q p ) for each p R. If (G A, D p ) has an nonuniform irreducible lattice, then for almost all p R f, D p is conjugate to SL n (Z p ) by an element of GL n (Q p ) Remark. We remark that the subgroup Γ need not be a lattice in G A 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), Λ S is an irreducible lattice in G S for any finite set S containing such that p S rank Q p (G p ) 2. Indeed, Λ S is a discrete subgroup of G S such that G(Q) +S Λ S G(Q) S. Note that G(Z S ) = G(Q) S and G(Q) +S is an infinite normal subgroup of G(Z S ). Hence by Margulis s normal subgroup theorem, G(Q) +S has finite index in G(Z S ) ([Ma2, Ch VIII, Theorem 2.6]). Therefore Λ S is a lattice in G S. From the assumption that G is absolutely simple, the subgroup G(Z S ) and hence Λ S is an irreducible lattices in G S. 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 H 1 (Q, G) is trivial, this happens for example if G = SL n. 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)) H 1 (Q, C). If G = PGL n and G = SL n, then H 1 (Q, C) = H 1 (Q, µ n ) = Q /(Q ) n where µ n is the n-th root of unity For a connected semisimple Q-group H, for almost all p R f, H is unramified over Q p, that is, quasi-split over Q p and split over an unramified extension of Q p. For such primes p R f, H(Z p ) is a hyperspecial subgroup of H(Q p ) or equivalently, a compact subgroup whose volume is maximum among all compact subgroups of H(Q p ). Hyperspecial subgroups of H(Q p ) are conjugate to each other by an element of H ad (Q p ) where H ad is the adjoint group of H [Ti, 3.8]. Theorem. Let T R and let G p be a connected semisimple adjoint Q p -group without any Q p -anisotropic factors for each p T. Assume that p T rank Q p (G p ) 2. Then

21 ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM 21 the group (G T, D p ) admits an irreducible lattice if and only if there exist a connected semisimple Q-simple Q-group H such that G p is Q p -isomorphic to a connected normal Q p -subgroup of H for each p T and D p is a subgroup whose volume is maximum among all compact subgroups of G p (Q p ) 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 f p : H G p a Q p -epimorphism for each p T. Let S be a finite subset of R f such that for any p R f S, H unramified over Q p and H(Z p ) is a hyperspecial subgroup of H(Q p ). By the hypothesis on D p, we can find a hyperspecial subgroup D p H(Q p) such that D p f(d p ) for each p (R f S) T. Hence for each p (R f S) T, there exists g p H ad (Q p ) such that g p D p g 1 p = H(Z p ). Let φ p = intg p if p (R f S) T and φ p = id if p R {S, } T. Then φ = p T φ p yields a topological group isomorphism between (H T, D p) and (H T, H(Z p )). Since H(Q(T)) is an irreducible (H being almost Q-simple) lattice in (H T, H(Z p )), φ 1 (H(Q(T))) is an irreducible lattice in (H T, D p). For each p T, write H = Hp 1 kerf p and D p = M1 p M2 p so that M1 p H1 p (Q p) and Mp 2 kerf p(q p ). Since p T (H1 p (Q p) Mp 2) (H T,D p ) is an open subgroup of (H T, D p ), the intersection φ 1 (H(Q)) p T (H1 p(q p ) Mp) 2 is a lattice in p T (H1 p(q p ) Mp) 2 (H T, D p). Now the canonical projection pr : p T (H1 p (Q p) M 2 p ) (H T,D p ) (H1 T, M1 p ) has compact kernel, the subgroup pr(φ 1 (H(Q)) p T (H1 p(q p ) M 2 p)) is a lattice in (H 1 T, M1 p ). Since the restriction of p T f p provides a topological group isomorphism from (H 1 T, M1 p) onto (G T, D p ), we obtain a lattice in (G T, D p ). Since H is Q-simple, the lattice obtained this way is irreducible, otherwise, it would yield a proper Q-subgroup of H Corollary. Let H be a connected absolutely simple Q-group. Let D p H(Q p ) be a compact open subgroup for each p R f. If (H A, D p ) admits an irreducible lattice, then D p is conjugate to H(Z p ) for almost all p R f. 5. Discrete subgroups containing lattices in horospherical subgroups In the whole section 5, for each p R, let G p be a connected semisimple adjoint Q p - group without any Q p -anisotropic factors and D p a maximal compact open subgroup for almost all p R f. We will say that (G A, D p ) has a Q-form (resp. Q-isotropic form) if there exists a connected semisimple adjoint (resp. Q-isotropic) Q-group H and a Q p -isomorphism f p : H G p for each p R such that f p (D p ) = H(Z p ) for almost all p R f. If (G A, D p ) has a Q-form, we denote by G A (Q) (resp. G A (Q) + ) the image of H(Q) (resp. H(Q) + ) under the restriction of p R f p to (G A, D p ).