Nguyen, Q T

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Available at: http://www.ictp.trieste.it/~pub off IC/2000/117United Nations Educational Scientic and Cultural OrganizationandInternational Atomic Energy AgencyTHE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICSA NOTE ON ZARISKI DENSE SUBGROUPS OF SEMISIMPLEALGEBRAIC GROUPS WITH ISOMORPHIC p-ADIC CLOSURESNguye^~n Q. Thang1Institute of Mathematics, P.O. Box 631, Bo Ho, Hanoi, VietnamandThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.AbstractWe prove under certain natural conditions the niteness of the number of isomorphism classesof Zariski dense subgroups in semisimple groups with isomorphic p-adic closures.MIRAMARE { TRIESTEAugust 2000Introduction.The present paper was inspired by Mazur [Ma], where he considered various types of local-globalprinciples in number theory and also the problem, for a given number eld k, to determinethe companions of a given algebraic k-variety V (i.e. those k-forms of V , locally everywherekv-isomorphic to V ). He also conjectured that for projective smooth varieties V over k, thereare, up to k-isomorphism, only nite number of companions of V . For algebraic groups whichare not necessarily linear, such a (well-known) question was answered in armative by Boreland Serre [BS]. We consider here an analog in the case of Zariski dense subgroups of semisimplegroups. The following provides a connection with similar question. Let k be a number eld, S anite set of valuations of k, containing the set1 of archimedean ones. Let O = O(S) be the ringof S-integers of k,  be a xed universal domain containing k. For a valuation v of k we denoteby kvthe v-adic completion of k, Ov= v-adic integers of kv, mv= maximal ideal of Ov, A =adele ring of k. Algebraic groups under consideration are identied with their points over .Assume that G  G(k), G ,! GLn(), where G denotes the Zariski-closure of G in GLn, G(B)will denote the B-points of a linear algebraic group G, with respect to the matrix realization ofG ,! GLnand for some ring B. Clv(G) denotes the (v-adic) closure of G in G(kv) with respectto the v-adic topology on G(kv). So there attaches to a given G a collection (Clv(G))vof v-adicclosures of G, which measures how big G is locally. One may ask the following natural question:() To what extent the collection (Clv(G))vdetermines the group G up to isomorphism ? Isthe number of isomorphism classes nite ?We are most interested in the niteness aspect of above question, i.e., given topological iso-morphisms Clv(G) ' Clv(Gi) for all v, where i runs over a set of indices I, we ask whether theset of isomorphism classes of fGigiis nite.These questions are closely related also to the congruence subgroup problem and strongapproximation in simply connected algebraic groups in its wide sense.It is our objective to establish the niteness of the number of isomorphism classes in the caseof semisimple groups (a partial answer to ()).In general, this is a dicult question and we will show the niteness to hold under certainrestrictions. The rst restriction is to require the groups Gito be "big" in the sense below.For simplicity we restrict ourselves to the case k = Q. Let I be a set of indices. For eachi 2 I let Gibe a Zariski dense subgroup of simply connected absolutely almost simple Q-groupGi,! GLni, such that Gi Gi(Z) and Gi6' Gjif i 6= j. Assume that each Gisatises thefollowing condition\p(Gi(Q) \ Clp(Gi))) = Gi: (B)Here Clp() means taking the closure in the p-adic topology of Gi(Qp). This condition meansthat Giare "big" so that one can recover the group Gifrom local closures. A Zariski dense2subgroup Gi Gisatisfying this condition (B) such that all closures Clp(Gi) are open andcompact subgroups of Gi(Qp) will be called big.1 The TheoremOur main result can be stated as follows.Theorem. With the above notation and convention, let I be a set of indices and for i 2 I,Gibe a big subgroup of a simply connected absolutely almost simple Q-group Gi. Then thenumber of isomorphic classes of groups Giwith isomorphic p-adic closures is nite.The proof of the theorem will be given in few steps.We x two groups G, H from the set B(G) := fGigi2I. L(G) will denote the Lie algebraof a Lie (resp. v-adic or algebraic) group G and we x once for all a matrix realization of Ginto GLn(). Ad(G) will denote the adjoint group of G.1. Lemma. The set B(G) is a disjoint union of nitely many classes of groups GiwithQ-isomorphic Zariski closures.Proof. By our assumption, each p-adic closure Clp(Gi) is an open and compact subgroup ofGi(Qp) and they are isomorphic to each other as topological groups. Denote by fp: Clp(G) 'Clp(H) the given topological isomorphism, where G and H are two xed groups from fGigi.By [Pin], Corollary 0.3, fpcan be extended uniquely to a Qp-isomorphismfp: G ' H, so Gand H are Q-linear algebraic groups which are Qp-isomorphic for all p. By Borel - Serre [BS],Theoreme 7.1, it follows that such groups lie in nitely many Q-isomorphic classes.    From now on we assume that all groups Gihave Q-isomorphic Zariski closures.The following lemma shows the adele nature of the family (fp).2. Lemma. With notation as in the proof of Lemma 1, for almost all p,fpis a Zp-polynomialisomorphism with respect to the given matrix realization of the groups G and H.Proof. Recall that we have xed an embedding G  GLn(Q). Since fpis a topological iso-morphism, it is also an isomorphism of p-adic analytic Lie groups, thus it maps a open uniformpowerful subgroup SG(p-saturable subgroup in terminology of [La], or standard subgroup in ter-minology of [Se]) of Clp(G) onto a open uniform powerful subgroup SHof Clp(H) (see [DDMS],3Ch. 4, [Se], Ch. 4, [La] for more details). It follows from the denition of Lie algebras of ana-lytic groups ([Se], Ch. 5, [DDMS], Sec. 8.2, 10.4) and the construction of standard subgroupsthat L(SG) ' L(SH) as Zp-Lie algebras, i.e., with structural constants belonging to Zp(loc.cit.) so dfpmust be a Zp-linear map with repsect to a given matrix realization (which is alwaysxed). Since SGis an open uniform subgroup of Clp(G), its Lie algebra L(SG) is a Zp-latticeof L(Clp(G)), in particular, L(Clp(G)) = L(SG) ZpQp, and the same is true for H insteadof G. Therefore dfp: L(Clp(G)) ' L(Clp(H)) is dened over Zp, thus the same is true forisomorphism L(G) ' L(H), thus also for dp: Aut(L(G)) ' Aut(L(H)). Since the map dpisgiven by the following rule :dp:  7! dfp   df 1p;it follows thatf0p: Ad(G) ! Ad(H) will be a Zp-polynomial isomorphism. Since fpextendsuniquely to Qp-isomorphismfp: G ! H by [Pin], Corollary 0.3, the following diagram iscommutative :Gfp! H1??y2??yAd(G)f0p! Ad(H)here idenotes the corresponding isogeny. It follows that for those p not lying in the set Tof primes dividing m, where m = Card(Ker (1)),fpis also dened over Zp. Thereforefpisdened over Zpfor all p not belonging to T .We need the following lemma in the sequel in order to realise Aut(G) as linear algebraicgroup over Q.3. Lemma. With the above notation, let f1; : : : ; fNbe Q-rational functions over G whichare linearly independent over Q. Then there exists x1; : : : ; xN2 G(Q) such thatdet(fi(xj))1i;j;N2 Q n f0g:Proof. We prove by induction on N . The case N = 1 is trivial. Recall that G(Q) is Zariskidense in G. Denotef(x1; :::; xn) := det(fi(xj))1i;j;N:Let N > 1 and assume that we have found N , 1 points x1; : : : ; xN 1such thatc = det(fi(xj)1i;jN 1) 6= 0:Consider the following Q-rational function g(z) on G dened as followsg(z) := f(x1; :::; xN 1; z);4and expand the determinant g(z) after the last row we haveg(z) = a1f1(z) +   + aN 1fN 1(z) + cfN(z):If for all z 2 G(Q) we had g(z) = 0, then due to the Zariski density of G(Q) in G, it wouldfollow that g(z) = 0, hence c = 0 since f1; :::; fNare Q-linearly independent, which contradictsthe choice of c.Denote by M = Aut(G) the group of rational automorphisms of G. It is well-known thatM has a natural structure of linear Q-algebraic group (see, e.g., [BS], [HM]) We need a specicrealization of the group M , which plays a crucial role in our proof, as follows. Recall that Adenotes the adele ring of Q.4. Proposition. With the above notation there is a realization of M as a linear algebraicQ-group such that for every H 2 B(G) and for any Q-isomorphism g : H ! G, the family(g  fp) (p runs over all prime numbers) is belong to M(A).Proof. First we x a universal domain . It follows from results of [HM] that G is a conserva-tive Q-group, i.e., the group M acts locally nitely on the Q-algebra Q[G] of regular functionsdened over Q on G. As before, we x an embedding G ,! GLn() and let xij(1  i; j  n)be the coordinate matrix functions on G. Let V be the smallest nite dimensional Q-vectorsubspace of Q[G] containing xij; 1  i; j  n; which is M -invariant (i.e. V is generated byxijand their images under the action of M). Let ff1; :::; fNg be Q-regular functions over Gwhich form a Q-basis of V containing all xij(notice that all xijare Q-linearly independent). Bymultiplying fkwith a suitable integer, we may assume that fkare all Z-polynomial functions.For  2M let the action of  be given as follows : fi7! fi  = 1jNa()ijfj;where a()ij2  (=universal domain). Since the Q-basis ff1; :::; fNg contains all coordinatefunctions, it follows that the mapping :  7! (a()ij)is a faithful Q-representation of M into GL(V ), where the latter is identied with GLN()by means of the basis ff1; :::; fNg. Further we will identify M with a closed Q-subgroup ofGLN(). Thus 2M(Zp), a()ij2 Zp; 8i; j:Now letfp: G ' H5be the isomorphism extending the isomorphism fp: Clp(G) ' Clp(H) (sofpis dened over Qp)and let g : H ' G be any Q-isomorphism.We now choose elements x1; :::; xNas in Lemma 3. For the convenience, we denote aij= a(fp)ij,where p is xed. Then we have the following systems of equations(A1)8>>>>><>>>>>:f1(g fp(x1)) = a11f1(x1) +   + a1NfN(x1)f1(g fp(xN)) = a11f1(xN) +   + a1NfN(xN)(AN)8>>>><>>>>:fN(g fp(x1)) = aN1f1(x1) +   + aNNfN(x1)fN(g fp(xN)) = aN1f1(xN) +    + aNNfN(xN)Denoter = c=d := det(fi(xj))(1  i; j  N);where c; d 2 Znf0g. Since xi2 G(Q) are nite in number, we may assume that xi2 G(Z[S 11])for all i, where Z[S 11] is the localization at a nite set S1of primes, which contains the setof primes dividing c. By Lemma 2, for certain nite set S2of primes the isomorphismfp(seenotation above) is dened over Zpfor p 62 S2. For a nite set S3of primes, we see that g isdened over Zpfor p =2 S3. Let S = S1[ S2[ S3. Then by solving the system Atabove withrespect to at1; :::; atN, we haveaij= (1=r)dij;8i; j;where aij2 Zp[S 1]. So for p 62 S we have g  fp2M(Zp) as required.Denote byC(G) = f(fp) 2QpM(Qp) : fp(Clp(G)) = Clp(G);8p; andfp2M(Zp) for almost all pg .It is clear that C(G) is an innite subgroup ofM(A). Next we want to parametrize the set B(G)by assigning to each H 2 B(G) a double coset class in M(Q)nM(A)=C(G) dened as follows :If g : H ' G is a Q-isomorphism,fp: G ' H is the isomorphism extending fp: Clp(G) 'Clp(H) for all p, then we seta(G;H) :=M(Q)(g fp)C(G):6According to Proposition 4, (g fp) 2M(A) so a(G;H) is an element of the set of double cosetclasses M(Q)nM(A)=C(G).5. Proposition. The correspondence dened above is a well-dened map.Proof. First we have to show that the class M(Q)(g fp)C(G) does not depend on the choice ofg and (fp).Let g0: H ' G be another Q-isomorphism, f0p: Clp(G) ' Clp(H) be an isomorphism withthe extensionf0p: G! H for all p. Then we have() g fp= (g  g0 1)  (g0f0p)  (f0 1pfp):Since g  g0 1is a Q-isomorphism of G, g  g0 12M(Q). For all p we have(f0p 1fp)(Clp(G)) = Clp(G):Hence for all p we havef0p 1fp2M(Qp) and thus for almost all p,f0p 1fp2M(Zp), becausef0pandfpare so. Hence (f0p 1fp) 2 C(G). ThusM(Q)(g fp)C(G) =M(Q)(g0f0p)C(G):The injectivity of the map H 7! a(G;H) now follows from the following6. Proposition. If (G;H) and (G;K) have the same double coset class then H = K.Proof. With notation as in the proof of Proposition 5, by the assumption we have for allprimes pf fp= gQ(g  gp)hp;where gQ2M(Q) and (hp) 2 C(G). Denote f0= g 1Q f , g0p= gp hp. Then for all p we havef0fp= g  g0p;org 1 f0= g0pf 1p;i.e., g 1 f0is a Q-isomorphism H ' K, mapping Clp(H) onto Clp(K) for all primes p.For h 2 H  H(Q) we have (g 1f0)(h) 2 K(Q), and (g 1f0)(h) 2 Clp(K) for all p. Thus(g 1 f0)(h) 2 K(Q) \ (\pClp(K)) = Kby the assumption that the groups Giare big. Hence (g 1 f0)(H)  K: Similarly we have(f0 1 g)(K)  H;7i.e., (f 1 f0)(H) = K; and H ' K, hence H = K since all groups Giare mutually non-isomorphic. Proposition 6 is proved.Preceding observations show that the cardinality of B(G) is not greater than the cardinalityof M(Q)nM(A)=C(G). We want to show that the latter is nite. DeneD = D(G) := f(ap) 2 C(G) : ap2M(Zp) ;8pg;i.e., D = C(G)\M(A(1)), where A(1) denotes the subring of nite adeles of A. In particularwe haveCard(M(Q)nM(A)=C(G))  Card(M(Q)nM(A)=D:The following proposition plays a crucial role in the proof of the niteness of Card(M(Q)nM(A)=D).7. Proposition. There is only a nite number of subgroups of a given nite index m inG(Zp).Proof. Let R be a subgroup of index m in G(Zp). First we assume that R is a normalsubgroup. Then by considering the factor group G(Zp)=R we conclude that R contains thesubgroup G(Zp)mof G(Zp) generated by the m-powers. It suces then only to prove that[G(Zp) : G(Zp)m] <1:Passing to a uniform pro-p-subgroup G0of G(Zp) we need only show that G0mis of nite indexin G0. It is known ([DDMS], Theorem 3.16) that G0is a pro-p-group of nite rank, say, d,and G0is topologically generated by g1; :::; gd. Also, by (loc. cit., Theorem 4.9) there exists ahomeomorphism : Zdp' G0;such that (x1; :::; xd) = gx11   xgdd:Therefore  ((mZp)d) is an open subset of G0, since mZpis open in Zp. It is clear that ((mZp)d)  G0m; hence G0mis open in G0and also of nite index.Now we assume that R is not normal inG(Zp). Then it is well-known that R contains a sub-group R0normal in G(Zp) and of index [G(Zp) : R0] dividing m!, hence R0contains G(Zp)m!.Then the above proof shows that [G(Zp) : G(Zp)m!] <1, therefore the proposition follows.8. Remarks. The same proof of Proposition 7 gives the following (cf. also with [Seg]).a) For a given compact p-adic analytic group, the number of its subgroups of given index mis nite.8b) There is only a nite number of subgroups of G(Qp) containing G(Zp) with given index m.Now we denote byM(Zp; Clp(G)) := ff 2M(Zp) : f(Clp(G)) = Clp(G)g:From [MVW], Theorem 7.3, or [N], Theorem 5.4, we know that Clp(G) = G(Zp) for almostall p (say, for all p outside a nite set W of primes). By the choice of the functions fj(inthe proof of Proposition 4), they are Z-polynomial functions. So if f 2 M(Zp) then we havef(G(Zp)) = G(Zp). Hence for p 62W we haveM(Zp; Clp(G)) =M(Zp):We need also the following9. Proposition. M(Zp; Clp(G)) is of nite index in M(Zp).Proof. By assumption G  G(Z), so it follows that for all p we have Clp(G)  G(Zp) andit is a subgroup of nite index in G(Zp) since Clp(G) is an open subgroup of the compact groupG(Zp). Lett = [G(Zp) : Clp(G)] <1;and Clp(G) = A1; : : : ; Akbe all subgroups of G(Zp) of index t (see Prop. 7). Then we have forany f 2M(Zp)[G(Zp) : f(Aj)] = [f(G(Zp)) : f(Aj)] = [G(Zp) : Aj] = t;so f acts transitively on the set fA1; : : : ; Akg. Thus we obtain a homomorphism :M(Zp)! Sk;where Skdenotes the symmetric group on k symbols. Consequently we have[M(Zp) : Ker  ] <1:It is obvious that Ker  M(Zp; Clp(G)) and the proposition follows.Now we are able to show10. Proposition. With the above notation we haveCard(M(Q)nM(A)=D) <1:9Proof. We haveCard(M(Q)nM(A)=D) == Card(M(Q)nM(A)=(Qp62WM(Zp)Qp2WM(Zp; Clp(G)))) Card(M(Q)nM(A)=M(A(1))) Qp2W[M(Zp) :M(Zp; Clp(G))]<1by the main theorem of Borel [Bor] and by Proposition 9.Summing up all results above we have proved the following11. Theorem. Let I be a set of indices and for i 2 I, let Gibe a Zariski-dense subgroupof a simply connected absolutely almost simple Q-group Gi, such that Gi Gi(Z) are mutuallynon-isomorphic, but all their p-adic closures for all p are topologically isomorphic, and each Giis big in Gi. Then I is nite.Acknowledgements. I would like to thank Professor R. Pink for sending his papers whichhelped a great deal the work over this paper. This work has been done thanks to the supportby NCST, the Abdus Salam I.C.T.P and the Swedish International Developement CooperationAgency (S.I.D.A).References[Bor] A. Borel, Some niteness properties of adele groups over number elds, Pub. Math. I.H. E. S. v. 16 (1963), 101 - 126.[BS] A. Borel et J. -P. Serre, Theoremes de nitufe en cohomologie galoisienne, Comm.Math. Helv. v. 39 (1964), 111 - 164.[DDMS] J. D. Dixon, M.P.F. du Sautoy, A. Mann and D. Segal, Analytic pro-p-groups, LondonMath. Soc. Lec. Note Ser. 157, Cambridge, 1991; 2-nd ed., 1999.[HM] G. Hochschild and G. Mostow, Automorphisms of algebraic groups, J. Algebra v.23(1969), 435 - 443.[La] M. Lazard, Groupes analytiques p-adiques, Pub. Math. I. H. E. S. 26 (1965), 5 - 250.[MVW] C. R. Matthews, L. N. Vasserstein and B. Weisfeiler, Congruence properties of Zariski-dense subgroups I, Proc. London. Math. Soc. 48 (1984), 514 - 532.10[Ma] B. Mazur, On the passage from local to global in number theory, Bull. A. M. S. v. 29(1993), 14 - 50.[Pin] R. Pink, Compact subgroups of linear algebraic groups, J. Algebra, v. 206 (1998), 438- 504.[Se] J. -P. Serre, Lie algebras and Lie groups, Harvard Lecture Notes, 1964; (2nd printing: Lecture Notes in Math. 1500, Springer-Verlag, 1992.)[Seg] D. Segal, Some remarks on p-adic analytic groups, Bull. London Math. Soc. v. 31(1999), 149 - 153.11

A note on Zariski dense subgroups of semisimple algebraic groups with isomorphic $p$ -adic closures

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A note on Zariski dense subgroups of semisimple algebraic groups with isomorphic ppp-adic closures

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A note on Zariski dense subgroups of semisimple algebraic groups with isomorphic $p$ -adic closures