Journal of Lie Theory Volume 13 (2003) 13–20 ◯^{C}2003 Heldermann Verlag

Zariski Dense Subgroups of Semisimple Algebraic Groups with Isomorphic p-adic Closures

Nguyêñ Quôć Thǎńg^{∗}

Communicated by G. A. Margulis

Abstract. We prove under certain natural conditions the finiteness of the number of isomorphism classes of Zariski dense subgroups in semisimple groups with isomorphic p-adic closures.

1. Introduction.

The present paper was inspired by the work of Mazur [Ma], in which he considered various types of local-global priciples in number theory and also the problem, for a given number field k, to determine the companions of a given algebraic k-variety V (i.e., those k-forms of V , which are locally everywhere k_{v} -isomorphic to V ). He also conjectured that for projective smooth varieties V over k, there are, up to kisomorphism, only a finite number of companions of V . For algebraic groups which are not necessarily linear, the corresponding conjecture was confirmed by Borel and Serre [BS]. We consider here an analog in the case of Zariski dense subgroups of semisimple groups. The following remarks provide a connection with similar question. Let k be a number field, S a finite set of valuations of k, containing the set ∞ of archimedean ones. Let O = O(S) be the ring of S -integers of k, Ω be a fixed universal domain containing k. For a valuation v of k we denote by k_{v} the v-adic completion of k, and by O_{v} the ring of v-adic integers of k_{v}. The algebraic groups under consideration are identified with their points over Ω. Assume that G ⊂ G(O), G ↪ GL_{n}(Ω), where G denotes the Zariski-closure of G in GL_{n} and G(B) denotes the B-points of a linear algebraic group G, with respect to the above the matrix realization and for some ring B. Also, Cl_{v}(G) denotes the (v-adic) closure of G in G(O_{v}) with respect to the v-adic topology on G(k_{v}). So, attached to a given G, there is a collection (Cl_{v}(G))_{v} of v-adic closures of G, which measures how big G is locally. One may ask the following natural question: (∗) To what extent does the collection (Cl_{v}(G))_{v} determine the group G up to isomorphism? Failing that, is the number of isomorphism classes finite?

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag We are most interested in the finiteness aspect of above question, i.e., given topological isomorphisms Cl_{v}(G) ≃ Cl_{v}(G_{i}) for all v, where i runs over a set of indices I, we ask whether the set of isomorphism classes of {G_{i}}_{i∈I} is finite.

∗ Regular Associate of the Abdus Salam I.C.T.P.

These questions are closely related also to the congruence subgroup problem and strong approximation in simply connected algebraic groups in its wide sense.

It is our objective to establish the finiteness of the number of isomorphism classes in the case of Zariski-dense subgroups of almost simple simply connected groups, which are big in certain sense.

In general, this is a difficult question and we will show the finiteness to hold under certain restrictions. The first restriction is to require the groups G_{i} to be ”big” in the following sense. For simplicity we restrict ourselves to the case k = Q. Let G_{i} be a Zariski dense subgroup of a simply connected absolutely almost simple Q-group G_{i} ↪ GL_{n_{i}}, such that G_{i} ⊂ G_{i}(Z), for each i in certain set of indices I, and G_{i} ≄ G_{j} if i ≠ j. Assume that each G_{i} satisfies the condition

⋂_{p}(G_{i}(Q) ∩ Cl_{p}(G_{i}))) = G_{i}.  (B)

This condition means that the groups G_{i} are ”big” so that one can recover the group G_{i} from local closures. As it follows from Nori’s Theorem [No], these groups are arithmetic subgroups in G_{i}(Z). A Zariski dense subgroup G_{i} ⊂ G_{i} satisfying this condition (B) such that all closures Cl_{p}(G_{i}) are open and compact subgroups of G_{i}(Q_{p}) will be called big.

2. The Theorem.

Our main result can be stated as follows.

Theorem 2.1. Let G_{i} be Zariski-dense subgroups in simply connected absolutely almost simple Q-groups G_{i} (i ∈ I), such that G_{i} ⊂ G_{i}(Z), and that all G_{i} are big in G_{i}; assume further that they are mutually non-isomorphic, while their p-adic closures are topologically isomorphic for all primes p. Then the set {G_{i}}_{i∈I} is the disjoint union of a finite number of isomorphism classes. The proof of the theorem will be given in several steps.

We fix two groups G, H from the set B(G) := {G_{i}}_{i∈I}. The Lie algebra of a Lie (resp. p-adic or algebraic) group G will be denoted by L(G). We fix once for all a matrix realization of G into GL_{n}(Ω). The adjoint group of G will be denoted by Ad(G).

Lemma 2.2. The set B(G) is a disjoint union of finitely many classes of groups G_{i} with Q-isomorphic Zariski closures.

Proof. By our assumption, each p-adic closure Cl_{p}(G_{i}) is an open and compact subgroup of G_{i}(Q_{p}) and they are isomorphic to each other as topological groups. Denote by f_{p} : Cl_{p}(G) ≃ Cl_{p}(H) the given topological isomorphism, where G and H are two fixed groups from {G_{i}}_{i∈I}. By [Pi], Corollary 0.3, f_{p} can be extended uniquely to a Q_{p}-isomorphism f̄_{p} : G ≃ H, so G and H are Q-linear algebraic groups which are Q_{p}-isomorphic for all p. By Borel - Serre [BS], Théorème 7.1, it follows that such groups lie in finitely many Q-isomorphic classes.

From now on we assume that all groups G_{i} have Q-isomorphic Zariski closures. The following lemma shows the adele nature of the family (f_{p}).

Lemma 2.3. With the notation as in the proof of Lemma 2.2, for almost all p, f̄_{p} is a Z_{p}-polynomial isomorphism with respect to the given matrix realization of the groups G and H.

Proof. Recall that we have fixed an embedding G ⊂ GL_{n}(Q). Since f_{p} is an isomorphism of topological groups, it is also an isomorphism of p-adic analytic Lie groups (see e.g. [Se]. Ch. 5. Sec. 9, or [DDMS], Sec. 9.2), thus it maps a open standard subgroup S_{G} of Cl_{p}(G) onto a open standard subgroup S_{H} of Cl_{p}(H) (see [DDMS], Sec. 8. 4, [Se], Ch. 4, Sec. 8 for more details). From the definition of Lie algebras of standard groups ([Se], Ch. 5, Sec. 1, [DDMS], Sec. 4.5) and the construction of standard subgroups it follows that L(S_{G}) ≃ L(S_{H}) as Z_{p}-Lie algebras, i.e., as Lie algebras with structural constants belonging to Z_{p} ([Se], Ch. 5, Sec. 1) so that df_{p} must be a Z_{p}-linear map with respect to the given matrix realization (which is always fixed). Since S_{G} is an open uniform subgroup of Cl_{p}(G), its Lie algebra L(S_{G}) is a Z_{p}-lattice of L(Cl_{p}(G)) and in particular, L(Cl_{p}(G)) = L(S_{G})⊗_{Z_{p}} Q_{p} by definition ([DDMS], Sec. 9.5). The same is true for H instead of G. Therefore df_{p} : L(Cl_{p}(G)) ≃ L(Cl_{p}(H)) is defined over Z_{p}, thus the same is true for isomorphism L(G) ≃ L(H), and so also for d_{p} : Aut(L(G)) ≃ Aut(L(H)). Since the map d_{p} is given by d_{p}(ϕ) = df_{p} ∘ ϕ ∘ df_{p}^{−1}, it follows that f̄′_{p} : Ad(G) → Ad(H) will be a Z_{p}-polynomial isomorphism. Since f_{p} extends uniquely to Q_{p}-isomorphism f̄_{p} : G → H by [Pi], Corollary 0.3, the following diagram is commutative :

G ⟶^{f̄_{p}}  H
π_{1}↓  π_{2}↓
Ad(G) ⟶^{f̄′_{p}} Ad(H);

here π_{i} denotes the corresponding isogeny. It follows that for those p not lying in the set T of primes dividing m = Card(Ker (π_{1})), f̄_{p} is also defined over Z_{p}. Therefore f̄_{p} is defined over Z_{p} for all p not belonging to T.

In the sequel we need the following lemma in order to realize Aut(G) as linear algebraic group over Q.

Lemma 2.4. With above notation, let f_{1}, …, f_{N} be Q-rational functions over G which are linearly independent over Q. Then there exist x_{1}, …, x_{N} ∈ G(Q) such that det(f_{i}(x_{j}))_{1≤i,j,≤N} ∈ Q \ {0}.

Proof. We prove the claim by induction on N. The case N = 1 is trivial. Recall that G(Q) is Zariski dense in G. Define f(x_{1}, ..., x_{n}) := det(f_{i}(x_{j}))_{1≤i,j,≤N}. Let N > 1 and assume that we have found N − 1 points x_{1}, …, x_{N−1} such that c = det(f_{i}(x_{j})_{1≤i,j≤N−1}) ≠ 0. Consider the following Q-rational function g(z) on G defined by g(z) := f(x_{1}, ..., x_{N−1}, z), and expand the determinant g(z) by the last row. We obtain g(z) = a_{1}f_{1}(z) + ⋯ + a_{N−1}f_{N−1}(z) + cf_{N}(z). If for all x ∈ G(Q) we had g(x) = 0, then due to the Zariski density of G(Q) in G, it would follow that g(z) ≡ 0. Therefore c = 0 since f_{1}, ..., f_{N} are Q-linearly independent, which contradicts the choice of c.

Denote by M = Aut(G) the group of rational automorphisms of G. It is well-known that M has the natural structure of a linear Q-algebraic group (see, e.g., [BS], [HM]). We need a specific realization of the group M, which plays a crucial role in our proof, as follows. Let A be the adele ring of Q.

Proposition 2.5. With above notation there is a realization of M as a linear algebraic Q-group such that for every H ∈ B(G) and for any Q-isomorphism g : H → G, the family (g ∘ f_{p}), where p runs over all prime numbers, belongs to M(A).

Proof. First we fix a universal domain Ω. It follows from results of [HM] that G is a conservative Q-group, i.e., the group M acts locally finitely on the Q-algebra Q[G] of regular functions defined over Q on G. As before, we fix an embedding G ↪ GL_{n}(Ω) and let x_{ij}(1 ≤ i, j ≤ n) be the coordinate matrix functions on G. Let V be the smallest finite dimensional Q-vector subspace of Q[G] containing x_{ij}, 1 ≤ i, j ≤ n, which is M -invariant (i.e. V is generated by x_{ij} and their images under the action of M). Let {f_{1}, ..., f_{N}} be Q-regular functions over G which form a Q-basis of V containing all x_{ij} (notice that all x_{ij} are Q-linearly independent). By multiplying f_{k} with a suitable integer, we may assume that all f_{k} are Z-polynomial functions.

For ϕ ∈ M let the action of ϕ be given by ϕ(f_{i}) = f_{i} ∘ ϕ = Σ_{1≤j≤N}a_{ij}^{(ϕ)}f_{j}, where a_{ij}^{(ϕ)} ∈ Ω (=universal domain). Since the Q-basis {f_{1}, ..., f_{N}} contains all coordinate functions, it follows that the mapping Φ : ϕ ↦ (a_{ij}^{(ϕ)}) is a faithful Q-representation of M into GL(V), where the latter is identified with GL_{N}(Ω) by means of the basis {f_{1}, ..., f_{N}}. Further we will identify M with a closed Qsubgroup of GL_{N}(Ω). Thus ϕ ∈ M(Z_{p}) if and only if a_{ij}^{(ϕ)} ∈ Z_{p}, for all i, j. Now let f̄_{p} : G ≃ H be the (unique) isomorphism extending the isomorphism f_{p} : Cl_{p}(G) ≃ Cl_{p}(H) (so that f̄_{p} is defined over Q_{p} ) and let g : H ≃ G be any Q-isomorphism.

We now choose elements x_{1}, ..., x_{N} as in Lemma 2.4. For convenience we write a_{ij} = a_{ij}^{(f̄_{p})}, where p is fixed. Then we have the following systems of equations

(A_{1})  { f_{1}(g ∘ f̄_{p}(x_{1})) = a_{11}f_{1}(x_{1}) + ⋯ + a_{1N}f_{N}(x_{1}) ; ⋮ ; f_{1}(g ∘ f̄_{p}(x_{N})) = a_{11}f_{1}(x_{N}) + ⋯ + a_{1N}f_{N}(x_{N})

⋮  ⋮
(A_{N})  { f_{N}(g ∘ f̄_{p}(x_{1})) = a_{N1}f_{1}(x_{1}) + ⋯ + a_{NN}f_{N}(x_{1}) ; ⋮ ; f_{N}(g ∘ f̄_{p}(x_{N})) = a_{N1}f_{1}(x_{N}) + ⋯ + a_{NN}f_{N}(x_{N})

Write

r = c/d := det(f_{i}(x_{j})) (1 ≤ i, j ≤ N),

where c, d ∈ Z \ {0}. Since the elements x_{i} ∈ G(Q) are finite in number, we may assume that x_{i} ∈ G(Z[S_{1}^{−1}]) for all i, where Z[S_{1}^{−1}] is the localization at a finite set S_{1} of primes, which contains the set of primes dividing c. By Lemma 2.3, for certain finite set S_{2} of primes, the isomorphism f̄_{p} (see notation above) is defined over Z_{p} for p ∉ S_{2}. For a finite set S_{3} of primes, we see that g is defined over Z_{p} for p ∉ S_{3}. Let S = S_{1} ∪ S_{2} ∪ S_{3}. Then by solving the system A_{t} above with respect to a_{t1}, ..., a_{tN}, we have a_{ij} = (1/r)d_{ij}, for all i, j, where a_{ij} ∈ Z_{p}[S^{−1}]. So for p ∉ S we have g ∘ f_{p} ∈ M(Z_{p}) as required.

Denote by C(G) the set

{(f_{p}) ∈ ∏_{p} M(Q_{p}) : f_{p}(Cl_{p}(G)) = Cl_{p}(G), ∀p, f_{p} ∈ M(Z_{p}) for almost all p} . It is clear that C(G) is an infinite subgroup of M(A). Next we want to parametrize the set B(G) by assigning to each H ∈ B(G) a double coset class in M(Q)\M(A)/C(G) defined as follows :

If g : H ≃ G is a Q-isomorphism and f̄_{p} : G ≃ H is the isomorphism extending f_{p} : Cl_{p}(G) ≃ Cl_{p}(H) for all p, then we set a(G, H) := M(Q)(g ∘ f̄_{p})C(G). According to Proposition 2.5, (g ∘ f̄_{p}) ∈ M(A) so a(G, H) is an element of the set of double coset classes M(Q)\M(A)/C(G).

Proposition 2.6. The correspondence defined above is a well-defined map.

Proof. First we have to show that the class M(Q)(g∘f̄_{p})C(G) does not depend on the choice of g and (f̄_{p}).

Let g′ : H ≃ G be another Q-isomorphism, and for all p, let f′_{p} : Cl_{p}(G) ≃ Cl_{p}(H) be an isomorphism with the extension f̄′_{p} : G → H. Then we have

(∗)  g ∘ f̄_{p} = (g ∘ g^{′−1}) ∘ (g′ ∘ f̄′_{p}) ∘ (f̄_{p}^{′−1} ∘ f̄_{p}).

Since g ∘ g^{′−1} is a Q-isomorphism of G, g ∘ g^{′−1} ∈ M(Q). For all p we have

(f̄′_{p}^{−1} ∘ f̄_{p})(Cl_{p}(G)) = Cl_{p}(G).

Hence for all p we have f̄′_{p}^{−1} ∘ f̄_{p} ∈ M(Q_{p}) and thus for almost all p, f̄′_{p}^{−1} ∘ f̄_{p} ∈ M(Z_{p}), because f̄′_{p} and f̄_{p} are so. Hence (f̄′_{p}^{−1} ∘ f̄_{p}) ∈ C(G). Thus

M(Q)(g ∘ f̄_{p})C(G) = M(Q)(g′ ∘ f̄′_{p})C(G).

The injectivity of the map H ↦ a(G, H) now follows from the following

Proposition 2.7. If (G, H) and (G, K) have the same double coset class, then H = K.

Proof. With notation of the proof of Proposition 2.6, by the assumption we have for all primes p

f ∘ f̄_{p} = g_{Q}(g ∘ ḡ_{p})h_{p},

where g_{Q} ∈ M(Q) and (h_{p}) ∈ C(G). Write f′ = g_{Q}^{−1} ∘f, ḡ′_{p} = ḡ_{p} ∘h_{p}. Then for all p we have f′ ∘ f̄_{p} = g ∘ ḡ′_{p}, or g^{−1} ∘ f′ = ḡ′_{p} ∘ f̄_{p}^{−1}, i.e., g^{−1} ∘ f′ is a Q-isomorphism H ≃ K, mapping Cl_{p}(H) onto Cl_{p}(K) for all primes p.

For h ∈ H ⊂ H(Q) we have (g^{−1} ∘ f′)(h) ∈ K(Q), and (g^{−1} ∘ f′)(h) ∈ Cl_{p}(K) for all p. Thus

(g^{−1} ∘ f′)(h) ∈ K(Q) ∩ (⋂_{p} Cl_{p}(K)) = K

by the assumption that the groups G_{i} are big. Hence (g^{−1} ∘f′)(H) ⊂ K. Similarly we have

(f^{′−1} ∘ g)(K) ⊂ H,

i.e., (f^{−1}∘f′)(H) = K, and H ≃ K; thus H = K since all groups G_{i} are mutually non-isomorphic. Proposition 2.7 is proved.

The preceding observations show that the cardinality of B(G) is not greater than the cardinality of M(Q)\M(A)/C(G). We want to show that the latter is finite. Define

D = D(G) := {(a_{p}) ∈ C(G) : a_{p} ∈ M(Z_{p}) , ∀p},

i.e, D = C(G) ∩ M(A(∞)), where A(∞) denotes the subring of finite adeles of A. In particular we have

Card(M(Q)\M(A)/C(G)) ≤ Card(M(Q)\M(A)/D.

The following proposition plays a crucial role in the proof of the finiteness of Card(M(Q)\M(A)/D).

Proposition 2.8. There is only a finite number of subgroups of a given finite index m in G(Z_{p}).

Proof. Let R be a subgroup of index m in G(Z_{p}). First we assume that R is a normal subgroup. Then by considering the factor group G(Z_{p})/R we conclude that R contains the subgroup G(Z_{p})^{m} of G(Z_{p}) generated by the m-powers. Then it suffices only to prove that

[G(Z_{p}) : G(Z_{p})^{m}] < ∞.

Passing to a open standard subgroup G′ (of finite index) of G(Z_{p}) we need only show that G^{′m} is of finite index in G′. It is known ([DDMS], Theorem. 8.31), that G′ is a uniform pro-p-group of finite rank, say, d, and G′ is topologically generated by its d elements g_{1}, ..., g_{d} (loc. cit., Theorem 3.17). Also, by (loc. cit., Theorem 4.9) there exists a homeomorphism ψ : Z_{p}^{d} ≃ G′, such that ψ(x_{1}, ..., x_{d}) = g_{1}^{x_{1}} ⋯ x_{d}^{g_{d}}. Therefore ψ((mZ_{p})^{d}) is an open subset of G′, since mZ_{p} is open in Z_{p}. It is clear that ψ((mZ_{p})^{d}) ⊂ G^{′m}, hence G^{′m} is open in G′ and also of finite index.

Now we assume that R is not normal in G(Z_{p}). Then it is well-known that R contains a subgroup R_{0} normal in G(Z_{p}) and of index [G(Z_{p}) : R_{0}] dividing m!, hence R_{0} contains G(Z_{p})^{m!}. Then the above proof shows that [G(Z_{p}) : G(Z_{p})^{m!}] < ∞, therefore the proposition follows.

Remark 2.9. We can use similar arguments as in the proof of Proposition 2.8 to prove (compare also with [Seg]) that for a given compact p-adic analytic group, the number of its subgroups of given index m is finite. By using this, in combination with Bruhat - Tits result about maximal compact subgroups of reductive p-adic groups [BrT], one can show that there is only a finite number of subgroups of G(Q_{p}) containing G(Z_{p}) with given index m, up to G(Q_{p})- conjugacy. Now we denote by

M(Z_{p}, Cl_{p}(G)) := {f ∈ M(Z_{p}) : f(Cl_{p}(G)) = Cl_{p}(G)}.

From [MVW], Theorem 7.3, or [N], Theorem 5.4, we know that Cl_{p}(G) = G(Z_{p}) for almost all p (say, for all p outside a finite set W of primes). By the choice of the functions f_{j} (in the proof of Proposition 2.5), they are Z-polynomial functions. So if f ∈ M(Z_{p}) then we have f(G(Z_{p})) = G(Z_{p}). Hence for p ∉ W we have M(Z_{p}, Cl_{p}(G)) = M(Z_{p}). We also need the following

Proposition 2.10. M(Z_{p}, Cl_{p}(G)) is of finite index in M(Z_{p}).

Proof. By assumption G ⊂ G(Z); so it follows that for all p we have Cl_{p}(G) ⊂ G(Z_{p}) and Cl_{p}(G) is a subgroup of finite index in G(Z_{p}) since it is an open subgroup of the compact group G(Z_{p}). Let

t = [G(Z_{p}) : Cl_{p}(G)] < ∞,

and Cl_{p}(G) = A_{1}, …, A_{k} be all subgroups of G(Z_{p}) of index t (see Prop. 2.8). Then for any f ∈ M(Z_{p}) we have [G(Z_{p}) : f(A_{j})] = [f(G(Z_{p})) : f(A_{j})] = [G(Z_{p}) : A_{j}] = t. So f acts transitively on the set {A_{1}, …, A_{k}}. Thus we obtain a homomorphism ψ : M(Z_{p}) → S_{k}, where S_{k} denotes the symmetric group on k symbols. Consequently we have [M(Z_{p}) : Ker ψ] < ∞. It is obvious that Ker ψ ⊂ M(Z_{p}, Cl_{p}(G)) and the proposition follows.

Now we are able to show

Proposition 2.11. With above notation we have

Card(M(Q)\M(A)/D) < ∞.

Proof. We have Card(M(Q)\M(A)/D) = Card(M(Q)\M(A)/(∏_{p∉W} M(Z_{p}) × ∏_{p∈W} M(Z_{p}, Cl_{p}(G)))) ≤ Card(M(Q)\M(A)/M(A(∞))) × ∏_{p∈W} [M(Z_{p}) : M(Z_{p}, Cl_{p}(G))] < ∞ by the main theorem of Borel ([Bor]) and by Proposition 2.10.

The proof of Theorem 2.1 now follows from the results above. Acknowledgement. I would like to thank Professor B. Mazur for his interest and valuabale suggestions regarding this paper, Professor R. Pink for sending his papers which help a great deal the work over this paper, and Professor Hofmann for valuable suggestions toward improving the readability of the text. This work has been done thanks to the support of the Fund. Research Program of Vietnam, the Abdus Salam I.C.T.P (Italy), and the Swedish International Developement Agency (S.I.D.A).

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Nguyêñ Q. Thǎńg Institute of Mathematics P. O. Box 631, Bo Ho Hanoi-Vietnam nqthang@thevinh.ncst.ac.vn

Received February 13, 2001 and in final form May 31, 2002