A Finiteness Theorem for K2of a Number Field

Annals of Mathematics

A Finiteness Theorem for K2 of a Number FieldAuthor(s): Howard GarlandSource: Annals of Mathematics, Second Series, Vol. 94, No. 3 (Nov., 1971), pp. 534-548Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1970769 .

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A finiteness theorem for K2 of a number field

By HOWARD GARLAND

0. Introduction

The purpose of this note is to prove a finiteness theorem for K2 of a finite algebraic extension of the rationals. The methods are transcendental, utili- zing results from differential geometry and analysis on symmetric spaces, but the final result is number theoretic. However, the connection between K2 and such transcendental methods, especially the theory of Eisenstein series, is probably quite natural (see for example, Kubota [8]).

To begin with, we state the theorem, and toward this end, introduce some notation. Let Z. Q. R, and C denote respectively, the ring of integers, the field of rational, real, and complex numbers. Let k denote a finite algebraic extension of Q and J c k the ring of integers in k. For each place v of k, we let kv denote the completion of k at v. We say v is infinite if kv is isomorphic to R or C, and finite otherwise. When v is finite we let J, be the topological closure of J in kv. We let A, Af, lA.. denote the collections of all places, finite places, infinite places of k, respectively.

We let Ak =]Jj(Jv) kv , Af = v(J )

vetA veAf

where the products are the restricted direct products with respect to the subring Jv.

For a commutative ring R, SLn(R) will denote the group of all n x n matrices with coefficients in R and determinant one. If R is a locally compact topological ring, then SLn(R) has an induced topology, and with respect to this topology SLn(R) is a locally compact group. Now let G be a locally compact topological group and let S' denote the unit circle. Following Moore (see [121, [13]) we consider Hi (G, S'), the ith Eilenberg-MacLane group of G based on Borel cochains, and with respect to the trivial action of G on S' (also we note that [12] and [13] serve as references for all facts used here concerning these Borel cohomology groups). In particular, if G is discrete, the HI(G, S') are the usual Eilenberg-MacLane groups of G. We give SLn(k) the discrete topology (indeed k is a discrete subgroup of Ak when k is diagonally imbedded).



A FINITENESS THEOREM FOR K2 535

The diagonal imbeddings of k into Ak and Af induce corresponding imbeddings of SLn(k) into SLn(Ak) and SLn(Af), respectively. We can now state our finiteness theorem:

THEOREM 0.1. The cokernal of the restriction homomorphism

P,: HB2(SL,(Af)g S ) >H 2(SL,(k), S ) is finite for n ? 3.

From the results in Moore [13], and Matsumoto [9], we have for m > n, a commutative diagram

HB2(SLn(Akf) S)n ) H2(SL,(k), S ) (0.2)

2B(SL..(Akf)9 S) n) H2(SL,,(k), S )

where all the maps are restriction homomorphisms, and for n > 3, the vertical maps are isomorphisms. Hence, to prove Theorem 0.1, it suffices to prove it for some n ? 3. Furthermore, we remark that the above theorem is usually stated in "dual form"; i.e. each of the cohomology groups HB2(SLn(Af), S'), H2(SL,(k), S') has a structure of a locally compact abelian group, and thus to p. there corresponds an adjoint map of Pontrjagin duals. However, the Pontrjagin dual of H2(SL,(k), S') is just K2(k) = Hll(SLn(k)), n > 3 (where Hll(SL,(k)) denotes the fundamental group in the sense of Moore [13]) and the dual of HB(SL.(Af), S') is just (D, Af K2?P(kJ)9 by results of Moore [13] and Matsumoto [9] (see Lemma 2.3, Th. 12.2 in Moore [13]). Hence the above theorem says that the kernel of the map

(0.3) K2(k) - evelf K20P(k,)

is finite. This extends (and essentially depends upon) a result of Bass and Tate which says the kernel of (0.3) is finitely generated. Moreover, Theorem 0.1 has been proved by A. Brumer in the case when k is a totally real, abelian extension of Q.

Our first task will be to reduce the proof of Theorem 0.1 to a question concerning the cohomology of arithmetic groups. This reduction is due to Bass but since his argument is fairly short, we include it here. The finite generation of the kernel of the map (0.3) already follows from this reduction. Both Bass's reduction and the solution of the question on arithmetic groups which subsequently arises, can be carried out quite generally, so that Theorem 0.1 is really a special case of a theorem on a certain class of semi-simple algebraic groups defined over Q. But it must be added that in some cases, the symplectic case for example, our method only yields an upper bound on the



536 HOWARD GARLAND

rank of the kernel of the map corresponding to (0.3). Recently, A. Borel proved that these upper bounds in fact yield the rank of the kernel. More generally let G be a semi-simple, linear, algebraic Q-group, let G denote the R-rational points of G, and K c G a maximal compact subgroup. Let F be a torsion-free, arithmetic subgroup of G, and X the compact dual of the symmetric space X = K\G. Our argument shows that for a certain range of i (call this the admissible range), the ith betti number of X/F is at most equal to the ith betti number of X. Using a certain compactification with corners of X/F (see Borel and Serre [18]), Borel has now proved the reverse inequality, and hence equality (for certain i in the admissible range). As another applica- tion of the compactification with corners, he has obtained a simple proof for a square integrability criterion which is somewhat weaker than that in [4] (Borel's criterion already implies G = RkIQSLm, m > 6, satisfies condition (G, 2), below). We note that in the notation of [4], the manifolds Ko\P /Foo,q are just the corners of [18].

The paper is organized as follows: In ? 1 we derive Bass's reduction of Theorem 0.1 to a question concerning arithmetic groups. In ? 2 we consider this question about arithmetic groups, which is in fact the question of whether a certain Eilenberg-MacLane group of an arithmetic group is zero. We observe that this latter vanishing theorem reduces to two further questions: (a) a square integrability criterion for de Rham cohomology on certain non-compact Riemannian manifolds and (b) the boundary behavior at c- of certain harmonic forms on these manifolds. In ? 3 we settle these questions for the case necessary to prove Theorem 0.1 (the methods are quite general, however).

We take this opportunity to thank H. Bass for giving us a copy of his manuscript, to R. Langlands for showing us the relation between convolution and Stoke's Theorem, and to J. Milnor and J. Humphreys for helpful comments on our original manuscript.

1. Bass's Theorem

Let V be a finite-dimensional vector space over the complex numbers, with a k-structure. Let Vk be the space of k-rational points. Let G c Aut (V) be a linear algebraic group which is semi-simple and is defined over k, with respect to the given k-structure on V. Next we fix a basis a {e1, ..., (n = dim V) of Vk. By means of a, G is identified with a subgroup of GLn(C). Now for every field F containing k, we let GF denote the F-rational points of G. For a subring R c F, we let GR = GF n GLn(R). Finally, we let GAk and GAkf denote the adelization and finite adelization, respectively of G, and we set G = GR. For a set of places S of k we let G, =JveS Gks, and we set G,, = G,




when S is the set of infinite places of k. We recall that Gk is diagonally imbedded in GAk and GAkf and from Kneser [7], and Platonov [14], we have

THEOREM 1.1. (Kneser-Platonov): Let G be a linear algebraic group which is absolutely simple, and is defined over k. Assume that G is simply connected, and that Go. is not compact; then Gk is dense in GAkf.

For the proof, see Kneser [7] and Platonov [14]. We remark that if G is defined and simple over Q, then for a suitable algebraic extension k of Q, G = RkIQH, where H is absolutely simple and is defined over k. Thus if H satisfies strong approximation (i.e. if Hk is dense in HAkf) then so does G (i.e. GQ is dense in GAQf). From now on, we assume G is in fact defined and simple over Q, and satisfies strong approximation. We now introduce a topology on GQ associated to congruence subgroups.

Definition 1.2. For a positive integer m, Fm c Gz will denote the subgroup of all matrices in Gz which are congruent to the identity modulo m. F c GQ will be called a congruence subgroup in case F contains Fm for some positive integer m.

The fact that the definition of congruence subgroups does not depend on the choice of a basis a of VQ follows from

LEMMA 1.3. Let (D: G - SLN be a rational representation of G which is defined over Q; then there is a positive integer m such that <D(rFm . SL,(Z).

The proof is simple and can be found in Weil [17].

Definition 1.4. We define the congruence subgroup topology on GQ by taking the congruence subgroups as a neighborhood basis of the identity.

We observe that the congruence subgroup topology on GQ is the same as the topology induced on GQ by the finite adele topology on GAvf. This follows from the following characterization of congruence subgroups: For each rational prime p let Qp and Zp denote the p-adic rationals and p-adic integers, respectively. Let 6: GQ - GAQf denote the diagonal imbedding. Then F c GQ is a congruence subgroup if and only if for some open subgroup U of fiP GzPy F = &-(&(Gz) n u) (we note that llP Gz, is an open subgroup of GAQf). It follows that 3 induces a topological isomorphism of GQ, the completion of GQ with respect to the congruence subgroup topology, onto GQ, the closure of GQ in GAQf. But in the light of Theorem 1.1, we have

COROLLARY (to Th. 1.1). Let H be a linear algebraic group which is absolutely simple, is defined over k, and is simply connected. Let G = RkQ(H) and assume G is not compact. Then the diagonal imbedding 6: GQ - GAQf



538 HOWARD GARLAND

induces a topological isomorphism of GQ onto GAQf.

Definition 1.5. A subgroup r c GQ will be called arithmetic if r F Gz is of finite index in r and in Gz. By the arithmetic (subgroup) topology on GQ we will mean the topology with arithmetic subgroups as a neighborhood basis of e.

We let G4 denote the completion of GQ with respect to the arithmetic topology. We will presently assume

Condition 1.6. (a) The kernel C of G4 -- GQ is central. (b) C is finite, and 1 - C - G - -*) 1 is a topological central extension in the sense that c is a topological isomorphism of C onto a closed subgroup of GQ, and w induces a topological isomorphism GQ/e - GQ. The sequence is also a covering in the sense that [G4, G41 = G*.

Remark. If G = RkIQH, with H a simple, simply connected Chevalley group of rank >2, then (a) is satified by Matsumoto [9] and (b) by Moore [13, Th. 13.1].

We then have

THEOREM 1.7 (Bass). Let G = Rk/QH, where H is a simply connected, absolutely simple Chevalley group of rank at least two. Let X e H2(GQ, S') and assume X restricted to Gz is zero. Then there is a cohomology class

e HI(GAQf, S') such that X restricts to 0 on Gz, the closure of Gz in GAQf and X restricts to X on GQ.

Proof. Let

e - S' c, G - GQ - e

be a central extension corresponding to X. By our assumption that X restricted to Gz is zero, we can find a homomorphism v: Gz - G such that *oO = identity. We then define a topology on G as follows: We take as a neighborhood basis 9Z of the identity in G, the sets of the form Uv(r), with U a neighborhood of the identity in S1, and r an arithmetic group. Of course, we must verify that SZ is indeed a neighborhood basis for a topology, and this means we must check that SZ satisfies certain well known conditions (see Pontrjagin [15]). For most of these conditions, the checking is direct, but one does require some work; namely,

If Ve 9Z, a e G, then there exists V1 e SZ such that a-' V1a c V.

Let V = Ua(r), a e G, a' = A(a). Set A = Gz n a' Gza'-' and

A = (r n [A, A]) n a'(r n [A, A])a'-' .




We note that under the hypotheses of our theorem, [A, A], and hence a, is arithmetic (see for example, Kazhdan [6]. Also note that this is just the assertion that H'(A, R) = 0, and that such a vanishing theorem (in a large number of cases) also follows from our present methods). Set

f (y) = (acca(Y)a) a(a'`'a')-' e S', for - e A.

A direct computation shows f is a homomorphism from A to SI. If V1= Ua(A), we then have (since f restricted to [A, A] is trivial)

a-'Vla = Ucr'a(/)a = Ua(a'`Aa') C Ua(r) = v,

and hence 9R does define a topology ? on G. One can then check that the left and right uniformities for ? are the same. We form the completions G*, GQ, of G with the topology ?, and GQ with the arithmetic topology, respectively. We now verify that

(i) * induces a continuous, surjective homomorphism A*: G* G*, and the kernel of A* is SI.

(ii) e , S' C G* P G4 * e is a topological central extension.

Let GI denote the closure of Gz in G*. Then from the remark below Condition 1.6, and from our hypotheses, we can assume Condition 1.6 is satisfied. But Condition 1.6 implies G* is an open subgroup of G4 (since the closure of GZ is open in GQ = GAQf). The first assertion of (i) now follows, since the image of '* is both open and dense (the continuity is clear).

We now prove the second assertion of (i). Assume h e G* and +*(h) = e. Let hi e G be a sequence approaching h. Then *(hi) approaches e in GQ, and hence is eventually in Gz. Hence hi e S'u(GL) for i large enough. Thus, pas- sing to a suitable subsequence, we may assume

hi = zii9, zi e S', Si e a(Gz) a

where z -. z e S'. But then hz-' = limiti hizT l = limit1 ,i = e. It follows that e - S' - C- * - GP - e is a central extension.

e(S') = 4*l(e) and S' is compact, so t is a topological isomorphism of S' onto a closed subgroup of G*. a extends to a continuous map a*: G* a*(G*), and A restricted to u*(G*) and a* are inverses of one another. It follows that A is open and hence induces a topological isomorphism G*/Sl _ Gi. Hence

e ?S ,G* GQ iee

is a topological central extension. Let C' = +*l(e), and consider the exact sequence

e -p c- G* + GQ )e.



540 HOWARD GARLAND

We note that this is a central extension. For we have a commutative diagram:

e e

e - S' - -> GQ- e

\\ {I

GQ

e

and for g e G*, c e C', set f(g) = gcglc-'. A direct computation shows f is a homomorphism from G* to S' (f (g) e S' since **(f (g)) = e), and f(gs) = f(g), for s e S'. Hence f induces a homomorphism from G* to S'. But by Condition Q

1.6, [G4, G4] = G*, and hence f(g) = 1 for all g e P*. Hence C' is central in G*. It is then clear (since C is finite and A and w are open) that

e - l C' -* GQ e

is a topological central extension. Now C' S' x iF, where if is a finite group (this follows from the fact that S' is infinitely divisible). We then obtain a topological central extension

e -> S' G*I>F e-,

and the corresponding class 'X e HB(GQ, S') (GQ = GAQf) satisfies the conditions of the theorem. Q.E.D.

Now Theorem 1.7 applies to RkIQSLm, for m > 3. It then follows that the cokernel of Pm is a homomorphic image of a subgroup of H2(SLm(J), S'), where J denotes the integers in k. Thus, to prove Theorem 0.1, it suffices to prove H2(SLm(J), R) = 0, for some m > 3. For we then have an exact sequence

0 H2(SLm(J), S') 9 H3(SLm(J), Z) 9 H3(SLm(J), R).

H3(SLm(J), Z) is a finitely generated abelian group, and in fact is essentially the third cohomology group of a compact differentiable manifold with boundary (see Raghunathan [16]). The kernel of 92 is torsion and hence finite, so

H2(SLm(J), S') - Image q-, c kernel q-2




is finite. In fact we shall prove H2(SLm(J), R) 0, for m > 7.

2. A vanishing theorem for the cohomology of arithmetic groups

In this section G will denote a linear algebraic group which is defined and simple over Q. Gz will be defined as in ? 1. We observe

(2.1) A subgroup of finite index in SLm(J) is an arithmetic subgroup of RkIQSLm,. Also (RkIQSLm)R ILveA SLm(kv)

Later on (Assumption (G, i), below) we shall introduce a technical condition on the group G. In ? 3 we shall prove

THEOREM 2.2. Let G be a linear algebraic group which is defined and simple over Q. Assume that (i) G satisfies condition (G, 2), below, (ii) every R-simple component of G is of the type described in either Theorem 5.2 (1) or Theorem 5.3, of [5], and (iii) every R-simple component of G is non-Hermitian; then for every arithmetic subgroup F c GQ, we have that H2(F, R), the second Eilenberg-MacLane group of F with respect to trivial action on R, is zero.

Remark. Conditions (i), (ii), (iii) are all satisfied if G = Rk!QSLm, m > 7. We will check (i) at the end of ? 3. (ii) follows from [5], and (iii) is well known (SO(n) and SU(n), n > 7, certainly have discrete center).

In the remainder of this section we will give a geometric interpretation of the cohomology groups H~(F, R), and introduce the condition (G, i).

First we note that F is finitely generated (see Borel [1]) and hence by a result of Selberg (ibid) F contains a normal subgroup A of finite index in F so that A is torsion-free. Hence we have an injection

H'(F, R) > H'(A\, Ry'A ,

and thus, in order to prove Theorem 2.2, we can assume F is torsion-free. From now on, we assume

(2.3) F is torsion-free . Now let Kc G be a maximal compact subgroup. Then F acts (to the

right) on the symmetric space X = K\G, and by assumption (2.3), each v e r acts without fixed point on X. Hence X/F inherits the structure of a locally Riemannian symmetric space and since X is a cell, we have

H'(r, R)-- Hi(XIF, R) I

where H'(X/F, R) is the ith cohomology of X/F. Then by de Rham's theorem Hi(X/F, R) is isomorphic to the space of closed Coo i-forms on X/F modulo the exact ones. Let * denote the Hodge star operator on X (and X/F) determined by the Riemannian structure (which we choose explicitly in ? 3), so that if



542 HOWARD GARLAND

a) is an i-form on X/J7, then *a) is an n-r-i form, where n = dim G, r = dim K, so n - r = dim X. Given two Co i-forms w1, W2 on X/F, we set (a)1, 0)2) = ()1 A *a)2, when the integral is defined. We say a Co i-form a is square summable if (a, )) < oA. We can now state condition (G, i):

Condition (G, i): We say the algebraic Q-group G satisfies (G, i) if for every torsion-free arithmetic subgroup F c G and for every closed C- i-form w on X/F, we can find a Co i - 1 form X on X/F so that a - d\ (d = exterior derivative) is square summable.

Now assume (G, i) holds. From de Rham [2], we then know that every i form on X/r is cohomologous to a square-summable, harmonic i-form. A remark is in order here. Recall that on a Riemannian manifold we can define the formal adjoint a of d, and then the Laplacian A= d3 + ?d. There are, a priori, two ways to define what is meant by a harmonic form Ap. Namely, Acp = 0 or dcp = 8g = 0. Clearly the second condition implies the first. How- ever, if the Riemannian structure is complete and p is square summable it is known the two conditions are the same (this is a result of Andreotti- Vesentini- their proof is contained in [3]). To prove H'(r, R) = 0, it would suffice, when condition (G, i) is valid, to show that every square summable harmonic form is zero. If G is anisotropic over Q, then X/r is compact, and in this case such vanishing theorems have been proved in great generality (see Matsushima [10], [11] and Kaneyuki-Nagano [5]). In order to apply their computation to the isotropic case (K\G/lr not compact) we must have some information about the boundary behavior at oo of square summable harmonic forms on K\G/r. In the next section we describe exactly what condition is required on the boundary behavior at oo of square summable harmonic forms, and derive it for "sufficiently many" forms.

3. The boundary behavior at cc of square summable harmonic forms

We begin by giving a convenient description of the complex of differential forms on X/r and of the Laplacian mentioned in ? 2 (for a more detailed discussion of these matters, see Matsushima-Murakami [111). Let ( denote the Lie algebra of G, where we identify ( with the right invariant vector fields on G. We let A denote the subalgebra of ( corresponding to K. B(, ) will denote the Cartan-Killing form on (, and f3 will denote the orthogonal complement of A with respect to B( , ). As is well known B( , ) is positive- definite on 3 and negative-definite on R2. We let n = dim (, let Greek indices X, pu, v range from 1 to n, Greek indices a, A8, y range from 1 to r (r - dim K), and Latin indices i, j, k range from r + 1 to n. We choose a basis X2, X-




1, .* , nof isothattheXaareabasisofS9, theXiareabasisof I3, B(Xa, X,5)= -ban and B(Xi, Xi) = 8ij. $3 may be identified with the tangent space to X at {K}. B 1,, the restriction of B to $, is invariant under the adjoint action of K, and hence determines a right invariant Riemannian structure on X and then a quotient Riemannian structure on x/r (this is the Riemannian structure we referred to in ? 2).

Next we give the description of the complex of Co differential forms on X/r. To begin with we let co, X = 1, * * , n be a dual basis to the X2, so the co are right invariant one-forms on G. We identify the X, and co' with their corresponding vector fields and one-forms on G/r. Let $,3* be the span of the w00s and let at: K - Aut (A'83*) be the representation induced by the adjoint action of K on 1I3. We can identify the space of complex-valued Co i-forms on x/r with the space C1(G, r) of all C- forms

o ( oi*1 l A ... A (&)Al on G such that the fj1. i are complex valued functions (complex differential forms will yield the cohomology H'(r, C), but this is just H'(r, R) ORC) satisfying

fil- 41(g9) = fil,...l(g), g e G, ^t e r (3.1) ***, fi1... ,l(kg)cwjl A ...A COik

= ,'**~L j3* ft,,il(g)ul(k)("il A ... A oil) , k e K, g e G.

Now if f is a Co function on G, and Y is a right invariant vector field, then 8'( Y)f will denote the Lie derivative of f with respect to Y. We remark that Y is an infinitesimal left translation so if f is right F-invariant, then so is 8'(Y)f. Now 0' is a representation of a and hence induces a representation of U((O), the universal enveloping algebra of (X. We denote this induced representation again by 0'. The Casimir operator C = Xi -

E-a X2 is in the center 3(e) of U(s). If

Q = Ef6 . wA* A A Aoil G Cl(G, r) then we have the following formula of Kuga for the Laplacian A = d3 + 3d applied to Q:

(3.2) AQ = .ii (O'(C)f,,....1))iI A ... A coil

Finally, the Riemannian structure also defines an inner product between certain forms on x/r. In our present setup, we have that up to a positive multiple this inner product is just

(Q, A) = ti, GIP 1 111 A *tA (O

where



544 HOWARD GARLAND

Q = Efij,...,ij(0'1 A ***A coil,9 A = E hil,... Ce(il A ...A nOil

We let 51(G, r) be the subspace of all elements Q in 61(G, r) with (Q, Q) < Ao.

We are then led to study the following more general setup. Let V be a finite-dimensional, complex vector space with a fixed Hermitian inner product, and let p: K - Aut V be a unitary representation with respect to this Her- mitian structure. Let C(p) be the space of all Co, V-valued functions f on G such that

f(gy) = f(g), fl(kg) = p(k)f(g), g e G, eCr,keK.

Thus C1(G, r) = C(ul), in our present notation. If Y is a right invariant vector field and if f is a Coo, V-valued function

on G, then O'( Y)f is the V-valued function such that ao(6'( Y)f) = O'( Y)(Uof) for every complex linear functional a on V. Just as before 0' determines an action of U(O) on Co, V-valued functions on G. Since C is in the center of U(O), 0'(C) maps C(p) into itself. For a real number X, we let

C(p, X) = {If G C(p) I 0'(C)f = Xf}I Thus C(ul, 0), for example, is the space of harmonic i-forms. We define g(p) c C(p) to be the set of all f e C(p) such that

5Gl'(f(g) f(g))ajl A .. A an < C0

where for v, w e V, v * w denotes their inner product with respect to the given Hermitian structure on V. We let 5(p, x) = 5(p) n C(p, X). Thus 5(l) = 1(G, F) and 5(al, 0) is the space of square summable harmonic i-forms on X/r.

If f1, f2 are complex-valued functions on G/F, we let

( I f=) = | T20) A *** A wn Glr

when the integral is defined. We say a complex-valued function f is square summable if (f, f) < Ac. If h is a V-valued function on G (or on G/F) we let hi, *.., h, denote the components of h with respect to a basis s',***, sa of V, which we assume fixed once and for all.

Let CC (G) be the space of complex valued Co functions on G with compact support and let a e CC(G) satisfy

(3.3) a(kgk-l) = a(g), ke K, ge G.

If h e C(p) (resp. 5(p)5(p, x)), then a*h e C(p) (resp. 5(p), (p, x)), where (a*h)i = a* hi, i = 1, * i i, 1.

LEMMA 3.4. Let ul, u2 G U(O) and Xe 0, and let h' e 5(p). Then if h = a*hh', with a e Cc(G) satisfying (3.3), we then have for any pair of component




functions hl, h2 of h (relative to our fixed basis of V) that each of the functions 0'(ui)hi, 6'(Xui)hi (i = 1, 2) is square summable, and

(6'(Xu1)h1, O'(U2)h2) = - ('(u1)h,, 6'(Xu2)h2)

Proof. Our assumption implies

hi= oa*h, i =1,2

where h' is a component function of h', and is square summable. The first assertion is clear. We then choose a sequence of Co complex-valued functions with compact supports, 0, on Gir so that (h' - 0, h-S) -? 0 as v o>. We then have

(6'(Xu1)h1, 6'(u2)h2) = (6'(Xu1)h1, O'(U2)a*h') = Limnitr (6'(Xu,)hl, 6'(U2)a*05r) = -Limit, (6'(u)h1, 6'(X)6'(u2)a*0Sr) = - (6'(u)h1, 6'(X)6'(U2)a*hf)

= -(0'(u1)hl, 6'(X)6'(u2)h2) . Q.E.D.

Now let Q' eG (U2, 0) be a square summable harmonic 2-form and let a e Cc (G) satisfy (3.3). Then a * Qf = Q C 5(U2, 0) and hence, as we mentioned earlier, dQ = 3Q = 0, by a result of Andreotti-Vesentini (see [3]). Now comes the main point: Lemma 3.4 implies that we have for the coefficients of Q a certain kind of integration by parts, and it follows that arguments in Ma- tsushima [10], for the compact case, along with the algebraic arguments in Kaneyuki-Nagano [5], go over verbatim to Q. Hence by the results of [10] and [5], Q = 0, provided G has all R-simple components non-Hermitian and of the type described in either Theorem 5.2 (1), or Theorem 5.3 of [5]. But any element in 5(q2, 0) is the L2 limit of functions a*Q', Q' eG (U2, 0) and a e CC (G) satisfying (3.3). Hence 5(U2, 0) = 0, provided G has R-simple factors of the same type we just mentioned. Therefore in the light of the results in de Rham [2]:

THEOREM 3.5. Let G be an algebraic semi-simple group which is defined and simple over Q. Let r c GQ be a torsion-free arithmetic subgroup and K c G a maximal compact subgroup. Assume each of the R-simple factors of G is non-Hermitian and of the type described in either Theorem 5.2 (1) or Theorem 5.3 of [5]. Set X = K\G. Then if q is a square summable 2-form on X/F, q is cohomologous to zero.

Remark. Recall our earlier remark (after Theorem 2.2) which says that the above theorem applies to RkIQSLm, m ? 7.

Finally, to complete the proof of Theorem 0.1, we need only verify that



546 HOWARD GARLAND

RkIQSLm, m > 7, satisfies condition (G, 2). We first briefly describe the square integrability criterion in Garland-Hsiang [4]. Thus let G be an algebraic linear group which is defined and simple over Q. We fix a maximal Q-split torus QS of G, a maximal R-split torus RS of G and a maximal torus T of G, so that

QSCRSCT ,

and we choose compatible orders on the roots. Let QA - {a, . . ., a1}, 1 = dim QS, denote the simple Q-roots. Let $ denote

the roots with respect to T, and $D+, $D- the subsets of positive and negative roots in .P, respectively. For 0 c QA, let D+(0) denote the set of all positive roots, fi in $D+ so that fi restricted to QS is not a linear combination of the elements of 6. W will denote the Weyl group of G with respect to T. Let j be a non-negative integer. We set (for 0 c QA)

wo = {a e w j($-) n $D+ CP ?+(0)} w(j) = {a e wj (o-) n $+ has j elements} wo(j) = wo n w(j) -

From Garland-Hsiang [41, we have (0 = empty set):

LEMMA 3.7. G satisfies condition (G, 2) if for j = 0, 1, 2, for a e W5(j), and g = 1/2,Eite~,,+ f,

(3.8) gIQS 1x1at Xi > ?.

If G = RkIQSLm, then G (considered as an algebraic group over C) has a direct product decomposition

G= II1=1 GA I

where each G is isomorphic to SLm(C). But then corresponding to this decomposition of G, T has a direct product decomposition

T = f-= T,

where each TR is a maximal torus in GA. Let Ad: T )T denote the projection. Then every Ti-root a extends to a T-root, namely to aoak. We identify a with this extension. For each x, the order on the T-roots induces an order on the T -roots, and if A denotes the simple T-roots, and AR the simple Th-roots, then A = ; A,. Moreover, if g2 denotes one half the sum of the positive T -roots, then g = = gn.

Now a e Ujsk W(j) if and only if a can be written as a product of no more than k simple root reflections. It suffices, in order to verify condition (G, 2), to verify that for a C U S;2 W(j),




(3.9) ? a= Ah, te cps I,, CP > 0 . This follows from the facts that each ai C QA is the restriction to QS of some ,a e A, and that the restriction to QS of each fi e A is either zero or an element of QA. Now if a is the identity, then (3.9) is trivially true. If a = Sp, the simple root reflection with respect to fi e A, then f8 e A2 for some x, and

(3.10) = gA + 91,

If a = SpSp, then if e e A19, f' e z, X 7 X', we have

(3.11) g= g-' ? g? ? , = S, 2-2 =

Finally, if fi, fi' A li, then

(3.12) gG = g E + gp

From (3.10), (3.11), (3.12), we see that it suffices to check (3.9) for G SLm. If m > 6, then

(3.13) g =fle- a &, ap > 5/2 .

Now gSA =g$ 8

= g- A- , if &, A' are orthogonal (9 S)Sfi,

g - 2R&' - A, if Af, fi' are not orthogonal .

But then from (3.13), we see that (3.9) holds for each a s UjI2 W(j). The proof of Theorem 0.1 is complete.

CORNELL UNIVERSITY, ITHACA, N. Y.

BIBLIOGRAPHY

I 1] A. BOREL, Introduction aux groupes arithmetiques, Publ. de l'Institut de Math. de I'Univ. de Strasbourg XV, Hermann, Paris, 1969.

f 2] G. de RHAM, Varietes differentiables, Publ. de I'Institut de Math. de l'Univ. de Nancago III, Hermann, Paris, 1960.

? 3] H. GARLAND, A rigidity theorem for discrete subgroups, Trans. Amer. Math. Soc. 129 (1967), 1-25.

4] and W. C. HSIANG, A square integrability criterion for the cohomology of arithmetic groups, Proc. Nat. Acad. Sci. U.S.A. 59 (1968), 354-360.

15] S. KANEYUKI and T. NAGANO, Quadratic forms related to symmetric Riemannian spaces, Osaka Math. J. 14 (1962), 241-252.

6] D. A. KAZHDAN, Connection of the dual space of a group with the structure of its closed subgroups, Functional Anal. Appl. (A translation of Functional Anal. i. Prilozen.) 1 (1967), 63-65.

[7] M. KNESER, Strong Approximation, Proc. Symp. Pure Math. 9, 187-196, Amer. Math. Soc. Providence, R. I., 1966.

1 8] T. KUBOTA, three talks given at Stony Brook, 1969, mimeo notes from Amer. Math. Soc. prepared in connection with the 1969 Summer Institute on Number Theory.

f9] H. MATSUMOTO, Sur les sous-groupes arithmetiques des groupes semi-simple de'ployes,



548 HOWARD GARLAND

Ann. Scient. Ec. Norm. Sup. (4) 2 (1969), 1-62. [10] Y. MATSUSHIMA, On Betti numbers of compact, locally symmetric Riemannian manifolds,

Osaka Math. J. 14 (1962), 1-20. [11] and S. MURAKAMI, On vector bundle valued harmonic forms and automorphic

forms on symmetric riemannian manifolds, Ann. of Math. 78 (1963), 365-416. [12] C. C. MOORE, Extensions and low dimensional cohomology theory of locally compact

groups I and II, Trans. Amer. Math. Soc. 113 (1964), 40-63, 64-86. [13] , Group extensions of p-adic and adelic linear groups, Inst. des Hautes Etudes

Scient., no. 35 (1968). [14] V. P. PLATONOV, An approximation problem and Kneser-Tits conjecture for algebraic

groups, Izvestiya Akad. Nauk SSSR 33 (1969), 1211-1219, and 34 (1970), 775-777. [15] L. PONTRJAGIN, Topological Groups, Princeton Mathematical Series 2, Princeton University

Press, Princeton, 1946. [16] M. S. RAGHUNATHAN, A note on quotients of real algebraic groups by arithmetic sub-

groups, Invent. Math. 4 (1968), 318-335. [17] A. WEIL, Discontinuous subgroups of classical groups, Univ. of Chicago notes (mimeo),

1958. [18] A. BOREL and J-P. SERRE, Adjonction de coins aux espaces symetriques; applications 2

la cohomologie des groupes arithme'tiques, Comptes Rendus 271, Serie A, (1970), p. 1156.

(Received January 6, 1971)



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