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 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) = Qp∈S Gp(Qp). We show that if Pp∈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 describe discrete 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) ∈ Gp(Qp) | gp ∈ Dp for almost all p ∈ Rf }. pY∈R 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 Q πS denotes the natural projection (GA, Dp) → p∈S Gp(Qp). The following is a sample case of our main theorem:Q 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 (QHA,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. i (1) Since S ⊂ RK −T (H), H(Kv) is non-compact for each v ∈ S. If we denote by Gp the maximal connected normal Qp-subgroup of Gp without any Qp-anisotropic i i factors for each p ∈ R and let Dp = Dp ∩ Gp for each p ∈ Rf , then in the above i i definition the quotient (GA, Dp)/kerf is isomorphic to (GA, Dp). 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. i i (2) If R0 = {p ∈ R | Gp(Qp) is non-compact}, then (GA, Dp) is naturally identified i with the restricted topological product of the groups Gp(Qp), p ∈ R0 with respect i i i i Q to the subgroups Dp. If R0 is finite, then (GA, Dp) = p∈R0 Gp( p). In this case, the above definition of a rational lattice in (GA, DpQ) coincides with that of i Q an R0-arithmetic (usually referred to as “S-arithmetic”) lattice of p∈R0 Gp( p) given in [Ma2, Ch IX, 1.4]. Q (3) If Γ is an irreducible lattice in (GA, Dp), then pr(Γ) is an irreducible lattice in i i i i (GA, Dp) as well where pr denotes the natural projection (GA, Dp) → (GA, Dp). Then an irreducible lattice Γ in (GA, Dp) is rational if and only if pr(Γ) is a i i rational lattice in (GA, Dp) 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 rank Qp (Gp) ≥ 2, any irreducible lattice in (GA, Dp) is rational. P 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. 4 HEE OH 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 = P GLn for each p ∈ R. Then (GA, Dp) has an irreducible lattice if and only if Dp is conjugate to P GLn(Zp) for almost all p ∈ Rf . For n = 2, for each p ∈ Rf , there are two conjugacy classes of maximal compact open subgroups of P GL2(Qp), represented by P GL2(Zp) and by a b 0 1 L = h ∈ P GL (Z ) , i p pc d 2 p p 0 respectively. Note that in the Bruhat-Tits tree associated to P GL2(Qp), the con- jugacy class of P GL2(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 2 denote by µp a Haar measure of P GL2(Qp), then µp(Lp) = p+1 µp(P GL2(Zp)) a b because the common subgroup ∈ P GL (Z ) has index p +1 in pc d 2 p P GL2(Zp) while it has index 2 in Lp.