Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Cohomology of Arithmetic Groups
Armand Borei
This paper is concerned with the Eilenberg-MacLane cohomology groups
H'(r ; E) of an arithmetic or S-arithmetic subgroup T of a reductive affine algebraic
group G over a number field, with coefficients in a ^-module E. For the sake of
brevity we shall, however, unless otherwise stated, assume G to be connected,
semisimple and the groundfield to be Q. Then S is a finite set of rational primes.
We recall that a subgroup T of the group G(Q) of Q-rational points of G is arithmetic (resp. S-arithmetic) if, given a Q-embedding p : G -+ GLn9 the group p(T) is
commensurable with p(G) f| GLn (Z) (resp. p(G) f| GLn(Zs), where Z$ is the ring
of rational numbers whose denominator is a product of elements in S).
I. GENERAL COEFFICIENTS
1. In this section S isfixed,T is S-arithmetic, torsion-free. From G and S alone,
one can construct a contractible locally compact (j(Q)-space X$9 whose z'th integral
cohomology group with compact supports H*C(XS ; Z) is zero except in one dimension e = e(G9 S)9 where it is a free module Is, on which T operates properly and
freely, so that the quotient Xs/T is compact, triangulable. It follows that Z admits
a Z[r]-ftee finitely generated resolution (as was shown first by Raghunathan [27]
when /Ms arithmetic (i.e., S is empty) and by J.-P. Serre [28] in general); hence
H*(r ; E) isfinitelygenerated if E is (as a Z-module), the cohottiological dimension
cà(r) of r is finite, equal to e(G9 S), T is a duality group in the sense of BieriEckmann [1], and we have a canonical isomorphism
(1)
W(r ; E) - ^ H^iiT ;IS®E)
(i e Z)
[5], [7], Let / = rkQ G be the Q-rank of G (common dimension of its maximal
Q-split algebraic tori) and lp the Q-rank of G9 viewed as a group over the field
© 1975, Canadian Mathematical Congress
435
436
ARMAND BOREL
Qp ofp-Sidic numbers (pe S). The space Xs is the product of the manifold with
corners X with interior the symmetric space X of maximal compact subgroups of
G(R)9 constructed in [6] (which is just X if / = 0), by the Bruhat-Tits buildings Tp
of the groups G(QP). If S = 0 , then Xs/r = X/T is a compact manifold with
corners, hence triangulable. In the general case, triangulability follows from properties of the projection Xs/ r -> Tsj T (where Ts = II ^ s Tp) and from results of
F. E. A. Johnson [18].
REMARK. Even if G is not semisimple, any torsion-free arithmetic subgroup of
G is a duality group [5], [7]. For further examples of discrete subgroups of products
of reductive groups over local fields (not necessarily of characteristic zero) which
have a homological duality, see [5], [7].
2. In the general case, r possesses torsion-free subgroups of finite index. Their
common cohomological dimension is, by definition, the virtual cohomological
dimension vcd(/T) of r [28], which is then finite, equal to e(G, S). The groups
H{(r ; E) are also finitely generated if .Cis. The group r also acts properly on Xs with
a compact quotient (presumably triangulable, too, but the author is not aware of
any proof outside the torsion-free case), which implies that the finite subgroups of
r form finitely many conjugacy classes. Further cohomological information on r
can be extracted from those subgroups; we mention some examples.
2.1. The group r and its subgroups of finite index F satisfy the conditions under
which an Euler characteristic yfT') e Q, in the sense of C. T. C. Wall, can be
defined. It is equal to X(F) = S ( - 1)'" dim H<(F ; Q) if F is torsion-free and satisfies the condition %(F') = [F : F'] i(F) if F' c F c T7 and [Z7 : F'] < oo. This
already implies that [T7:/7']- %(r) e Zif F is torsion-free. K. Brown [8], however,
has given an expression for ^(T7) — X(r), involving the lattice of finite subgroups of
T7, which implies in particular that m • %(r) e Z as soon as the order of any finite
subgroup of r divides m. Now let k be a totally real number field, n its degree over
Q9 oT the ring of elements in k which are integral outside a given finite set T of
finite primes of k and P= SL2(oT). Let further £Ä>r be the function obtained from
Dedekind's zeta-function Çft of k by omitting the local factors associated to the
primes in T. Then, by a special case of a formula of G. Harder [14], if S ^ 0 , by
[28, 3.7] in eneral, we have
(1)
x(SL2(oT)) =
i:ktT(-l).
The results mentioned above then allow one to get estimates of the denominator
of the right-hand side using finite subgroups or subgroups of finite index of T7.
(See [28, 3.7], [8, § 4], where products of values Ç*|7<1 — 2/) (1 ^ i ^ n) are also
similarly related to the symplectic groups Sp2«(or)-)
2.2. Given a prime p, there is an algebra Ap over Z/pZ constructed from the
category of elementary commutative /^-subgroups of T7, and a homomorphism
H*(r9 Z/pZ) -> Ap whose kernel and cokernel are annihilated by some power of
the/?th power map [26, §§ 14, 15]. Thus, asymptotically, the cohomology mod p
is to some extent determined by the commutative elementary /^-subgroups of T7.
COHOMOLOGY OF ARITHMETIC GROUPS
437
2.3. Assume now G to be the algebraic group defined by the elements of norm
one in a division algebra D over Q9 and T7 to be arithmetic. Then, via the regular
representation, the finite subgroups of F (and even G(Q)) operate on Rn - 0 (n *=
[D:Q]) freely since D is a division algebra; hence their cohomology is periodic, of
period dividing n. From this, using the spectral sequences of equivariant cohomology theory, B.B. Venkov [29] has shown the same to hold for Z7, from a certain
dimension on. This is also true if /Ms ^-arithmetic; the argument is the same, with
X x Ts (notation of § 1) playing the role of X.
2.4. We note finally that an easy spectral sequence argument shows that if d is a
common multiple of the orders of the finite subgroups of T7, then it annihilates
W(r ; E) for i > vcd(r).
II. REAL OR COMPLEX COHOMOLOGY
3. From now on, E is the space of a finite-dimensional real or complex representation of G(R)9 and, unless otherwise stated, T7 is arithmetic. (In fact, the results
recalled in this section and their proofs are valid for any discrete subgroup of
G(R).) The group G(R) operates in a natural way on the space QX(E) of smooth Evalued differential forms on X9 and H*(r ; E) is canonically isomorphic to the
cohomology of the complex Qx(E)r of Ainvariant elements in QX(E). Fix a
maximal compact subgroup K of G(R). Using the canonical projection of G(R)/T
onto X/F one identifies in a well-known way Qx(E)r with a space of smooth
vector-valued functions on G(R)/r, and this yields in fact a natural isomorphism
of Qx(E)r with the cochain complex C*(g, £ ; !F ® E) of relative Lie algebra
cohomology, where, if E is real (resp. complex), J5" is the space of smooth real
(resp. complex) valued functions on G(R)jT9 g and I are the Lie algebras (resp. the
complexifications of the Lie algebras) of G(R) and K9 and fF ® E is viewed as a
g-module in the obvious way. Thus we get a canonical isomorphism
(1)
i/*(g, ! ; & ® E) — > H*(r ; E).
(For all this, see [23], [24]; the blanket assumption X/T compact made there is not
used for these general remarks. Actually, (1) is correct as stated if G(R) is connected, also a standing assumption in [23], [24], or if /Ms contained in the identity
component of G(R). Otherwise, the left-hand side has to be replaced by the invariants under a suitable finite group of automorphisms. We shall ignore this technicality.) Let now "T be a subspace of !F stable under K and g. The inclusion "K ®E ->
!F ® E then induces homomorphisms
(2)
p : #<g, I ; r ® E) -> H*(r ; E)
fe=0,
1, 2 , - )
which we shall discuss in three cases (in §§ 4, 6, 7).
4. Stable cohomology. Take E = f = JE, with the trivial action of G(R)9 where
°T is the space of real constant functions on G(R)/r. We have then a natural homomorphism/?: HQ(Q9 Ï;R)-+ Hv(r;R). If /Ms cocompact (i.e., G(R)/Tis compact),
then !F has a G(j?)-invariant supplement to T\ hence/« is injective for all q (as was
438
ARMAND BOREL
remarked first, I think, by W. T. van Est [9, Theorem 7], from a different point of
view) but this is not so in the noncocompact case. Moreover, again in the cocompact case, but without assuming Z7 to be arithmetic, Matsushima has shown j * to
be surjective at least up to a constant m(G) computable from g [21]. Similarly, in
the arithmetic case, we have the following:
THEOREM
4.1. The homomorphism fi is an isomorphism for q < (rkQG)/4.
(See [3], [4]; the surjectivity part had already been proved in substance by H.
Garland [11, 3.5] or [4, 3.5], and used by him to show that K2o is a torsion group.)
Let Gu be a maximal compact subgroup of G(C) containing K. Then Xu =
K°\GU)9 where K° is the identity component of K9 is a compact symmetric space,
the "dual" space to X. As is well known, and goes back to E. Cartan, 7/*(g, t ; R) is
canonically isomorphic to H*(XU ; R)9 whence a homomorphism a* : H<i(Xu ; R)
—> H*(T ; E)9 which is an isomorphism at least in degrees < (rkQG)j(p.
Let now (Gni Tn9 Xn, Xn>u) be objects similar to (G, Z7, X, Xu) and /„ : (Gni r„)
-> (Gn+Ì9 rn+i) an injective Q-morphism (n = 1, 2, •••)• Let Xu — inj lim X„tU.
Assume that rkQ Gn -• oo. Then, for many classical sequences of this type, in
particular the one of the next section, there exists, given q9 an integer n(q) such that
(1)
H*(XU\ R) S ff«(inj lim Tn ; R) = H*(XntH ; R) = H*(rn ; R)
(n*
n(q)).
(See [3], [4] for more precise statements, also pertaining to S-arithmetic groups.)
5. The cohomology of SL(o), higher regulators and values of ^-functions. We consider the special case where Gn = Rk/QSLni Fn — SLno (where k is a number
field, o its ring of integers, Rk/Q the restriction of scalars [30, Chapter 1]) and /„
comes from the natural inclusion SL„-> SLn+\. Passing to homology, we have then
an isomorphism a* :H*(SL(o);R) -> H*(XU\R)9 where SZ(o) = inj lim SLn(o).
For m — 1, 2, •••, let P2m+\ (resp. P2m+\) be the space of primitive elements in
I^2m+ì(SL(ó) ; R) (resp. H2m+1(Xu; R)). Its dimension dm is equal to r2 (resp.
r
\ + ri) if W7 ^ 1 is odd (resp. even), where /^ (resp. r2) is the number of real (resp.
complex) places of k; this also happens to be the order of the zero of Qk (s) at s =
— 777. There is a natural map of 7C2m+i(Xu) (resp. of Quillen's group K2m+io) into
Pzw+i (resp. i^w+i), whose image is a lattice I^m+1 (resp. £ 2w+ i). The mth regulator
of A: is then, by definition [20, § 4], the positive real number Rm such that the map
AdmP2m+i -+ Ad"P'Zm+i defined by a* sends Ad"L2m+i onto Rm-(AdL'2m+l).
Given two nonzero real numbers, write a ~ b if a/£ is rational.
THEOREM
5.1. Let Dk be the discriminant ofk over Q. For m ^ 1, we /?ave
i?w ~ J 9 p - 7 r w ( w + 1 ) ^ A ( m + 1).
According to the functional equation for ^(s), this is equivalent to
Rm - 7ü-d^lims^m^k(s)/(s
+ W7)rf".
This, however, says nothing about the actual quotient of these two numbers.
According to the conjectures in [20, § 4], slightly modified to take into account the
COHOMOLOGY OF ARITHMETIC GROUPS
439
factor 7c~dm, it should be closely related to the orders of K2mo and of the torsion
subgroup of K2m+io.
REMARK. Applied to other sequences of classical groups, the results of § 6 also
yield the ranks of Karoubi's groups eL^o or eLtk (see [3], [4]).
6. Cusp cohomology. We come back to the setting of § 3, assume E to be complex,
endowed with a hermitian scalar product invariant under K (or more precisely
admissible in the sense of [23]), andfixan invariant measure on G(R)/F This allows
one to define a scalar product on compactly supported elements of C*(g, t ; 3F ® E)
(and then on various bigger spaces), an adjoint 9 to d, and a Laplace operator
dd + dd [23], [24]. Let QL2(G(R)/r) be the space of cusp forms in the space
L2(G(R)/T) of complex-valued square-integrable functions on G(R) [17], [19]. By
a well-known result of Gel'fand and Piateckii-Shapiro [13], [17], it is a direct sum of
closed irreducible (under G(R)) subspaces, with finite multiplicities. Take now for
y the space °J* of C°°-vectors (in the sense of infinite-dimensional representation
theory) oî°L2(G(R)/r). Thus, *K consists of the elements fe & such that Xfvz
square-integrable for every Xe C/(g) and
f o r a11
Sum/mnfte'u) du==0
* e G(R)9
where U is the unipotent radical of an arbitrary proper parabolic Q subgroup of G.
THEOREM 6.1. The map °/* : #*(g, I ; °«T ® E) -> H*(r; E) is injective.
Its image will be denoted H*usp(r; E).
Let r be afinite-dimensionalrepresentation of K. For ce R9 let A(r9 c) be the
space of elements in L2(G(R)jT) which transform according to /* with respect to
left translations by K and are eigenfunctions of the Casimir operator with eigenvalue c. As is well known, these elements are in fact analytic. Let °A(r, c) =
A{r, c) R °L%G(K)/r).
Q
THEOREM 6.2. The space A(r9 c) isfinite-dimensional,contained in the sum of
finitely many closed irreducible subspaces of°L2(G(R)/r)'9 in particular, its elements
are Z-finite (Z center of £/(g)), and are automorphic forms in the sense of [17], [19].
Given r, the set of c for which °A(r9 c) is ^ {0} is bounded from above and has no
finite accumulation point.
(6.2, the results below, and those of §7 are joint work of H. Garland and the
author; they extend theorems proved by H. Garland for Q-rank one groups [10].)
In view of Kuga's formula relating the Laplace operator and the Casimir operator [23], it follows immediately from 6.2 that //*(g, I ; °& ® E) may be identified
with the space of harmonic forms in C*(g, t ; °^ ® E). Thus / ^ ^ ( Z 7 ; E) is isomorphic to the space of harmonic cusp forms. Assume now E to be irreducible. Let
c be such that the Casimir operator on E is c-ld. Write further °L2(G(R)ir) as a
Hilbert direct sum of irreducible subspaces H; and let c,- be the eigenvalue of C in
the space of differentiable vectors of Hi. Then we have
(1)
H^r
\E)^@
iyCi—C
Hom*(/«g/f) ® E9 Ht).
440
ARMAND BOREL
If is is the trivial representation, then c = 0. In the cocompact case (where ®@r=<F)
the formula overlaps with one of Matsushima's [22]. More generally, 6.2 implies
that a number of arguments in the cocompact case (such as the generalized EichlerShimura isomorphism in [24, § 7]) or the vanishing theorems in [24, § 11] remain
valid for cusp forms and cusp cohomology.
7. Square-integrable cohomology. Take now for Y the space of C°°-vectors in
L2(G(R)jr). The image of /* is then the space of cohomology classes which can be
represented by square-integrable forms (hence also by square-integrable harmonic
forms). It contains the image of the cohomology with compact supports, is equal
to it if G = Rk/Q SL2 by [15], but is bigger in general. However, 6.2, induction and
the results of § 7 in [19] allow one to prove that 6.2 remains true if °A(r, c) and °L2
are replaced by A(r9 c) and U (this was shown to us by Langlands). In particular,
a ^-finite eigenfunction of C in L2(G(R)/r) is Z-finite, contained in the sum of
finitely many elements of the discrete spectrum. This is the analogue of a result of
Okamoto for L2(G(R)) [25]. It also follows that the space of square-integrable Evalued harmonic forms isfinite-dimensional,isomorphic to Z/*(g, f ; y ® E) and
given by a formula similar to (1), but where °L2 is replaced by the discrete spectrum
in L2(G(R)/r)9 if E is irreducible.
8. The Q-rank one case. Assume now rkQ G = 1. The manifold with corners
V = X/r is then in fact a manifold with boundary, and the connected components
of its boundary dV correspond bijectively to the /'-conjugacy classes of minimal
parabolic Q-subgroups of G [2, § 17]. Letr :H*(V-9 E) -• H*(dV; E) be the restriction
homomorphism. Using Langlands' theory of Eisenstein series [17], [19], G. Harder
has shown the existence of a subspace H?n{ (r;E) of H*(T;E) s H*(V; E)9 which
restricts isomorphically onto Im r, whose elements are obtained either by taking
analytic continuation of suitable Eisenstein series, or residues of such at simple
poles [16]. Thus, in this case, every element of H*(r; E) has a closed harmonic
representative. If G = Rk/QSL29 and E = C9 G. Harder has given a complete
description of Im r; moreover, //*(Z7 ; C) is in this case the direct sum of Hfn{,
H*sp9 and the image of Z/*(g, f ; C) [15].
9. No such results have yet been obtained in the higher rank case. Still, they
point to extremely interesting relations between H*(Z7 ; E) and the theory of
automorphic forms, and lead one to wonder whether (a) all cohomology classes are
represented by closed harmonic forms; (b) there is a sum decomposition of
H*(T ; E)/H*usp(r ; E)9 where each summand is naturally associated to Eisenstein
series built from a class [P] of associated proper parabolic Q-subgroups, starting
from harmonic cusp forms on pieces of the boundary of the manifold with corners
X/r corresponding to the elements of {P}.
Finally, I would like to draw attention to two topics left out of this survey : vanishing theorems for subgroups of/?-adic groups and related questions, for which we
refer to H. Garland's article [12], and the use of cohomology of arithmetic groups
in the discussion of zeta-functions or L-functions of certain algebraic varieties
COHOMOLOGY OF ARITHMETIC GROUPS
441
(Eichler, Shimura and, more recently, I, I. Piateckii-Shapiro and R. P. Langlands
(in Modular functions of one variable. II, Springer Lecture Notes in Mathematics,
vol. 349) for modular curves; Shimura, Kuga-Shimura and Langlands for other
quotients of bounded symmetric domains),
References
1. R, Bieri and B. Eckmann, Groups with homological duality generalizing Poincaré duality9
Invent. Math. 20 (1973), 103-124,
2. A. Borei, Introduction aux groupes arithmétiques, Pubi. Inst. Math. Univ. Strasbourg, 15,
Actualités Sci. Indust., no. 1341, Hermann, Paris, 1969. MR 39 #5577.
3.
, Cohomologie réelle stable de groupes S-arithmétiques classiques, C, R. Acad. Sci.
Paris Sér. A-B 274 (1972), A1700-A1702. MR 46 # 7400.
4.
, Stable real cohomology of arithmetic groups', Ann. Sci. École Norm. Sup. (4) 7 (1974),
235-272.
5. A. Borei and J.-P. Serre, Cohomologie à supports compacts des immeubles de Bruhat-Tits;
applications à la cohomologie des groupes S-arithmétiques, C, R. Acad Sci, Paris Sér. A-B 272
(1971), A110-A113. MR43 #221.
6. •
, Corners and arithmetic groups, Comment. Math. Helv. 48 (1974), 244-297.
7.
, Cohomologie d'immeubles et de groupes S-arithmétiques (in preparation),
8. K, Brown, Euler-characteristics of discrete groups and G-spaces, Invent. Math. 27(1974),
229-264.
9. W. T. van Est, A generalization of the Cartan-Leray spectral sequence. I, II, Nederl. Akad.
Wetensch. Proc. Ser. A 61 = Indag. Math. 20 (1958), 399-413. MR 21 #2236.
10. H. Garland, The spectrum of noncompact G/T and the cohomology of arithmetic groups,
Bull. Amer. Math. Soc. 75 (1969), 807-811. MR 40 #269.
11.
, Afinitenesstheorem for K2 of a numberfield,Ann. of Math. (2) 94 (1971), 534-548.
MR 45 #6785.
12.
, On the cohomology of arithmetic groups, these PROCEEDINGS.
13. I. M. Gel'fand and I. I. Pjateckiï-Sapiro, Automorphic functions and representation theory,
Trudy Moskov. Mat. Obsc. 12 (1963), 389-412 =Trans. Moscow Math. Soc. 1963, 438-464. MR
28 #3115.
14. G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci.
École Norm. Sup, 4 (1971), 409-455. MR 46 # 8255.
15.
, On the cohomology ofSL(2, o) (preprint).
16.
, On the cohomology of discrete arithmetically defined groups, Proc. Internat.
Colloquium Bombay 1972 (to appear).
17. Harish-Chandra, Automorphic forms on semisimple Lie groups, Lecture Notes in Math,,
no. 62, Springer-Verlag, Berlin and New York, 1968. MR 38 # 1216.
18. F. E. A. Johnson, On the triangulation of stratified sets and singular varieties (to appear).
19. R, P. Langlands, On the functional equations satisfied by Eisenstein series, Proc. Sympos.
Pure Math., vol, 9, Amer. Math. Soc, Providence, R.I., 1965, pp. 235-252. MR 40 #2784.
20. S. Lichtenbaum, Values of zeta-functions, étale cohomology, and algebraic K-theory, Algebraic ^-Theory, II, Lecture Notes in Math., no. 342, Springer-Verlag, New York, 1973, pp.
489-499.
21. Y. Matsushima, On Betti numbers of compact, locally symmetric Riemannian manifolds9
Osaka Math. J. 14 (1962), 1-20. MR 25 #4549.
22.
, A formula for the Betti numbers of compact locally symmetric Riemannian manifolds,
J. Differential Geometry 1 (1967), 99-109. MR 36 # 5958.
23. Y, Matsushima and S. Murakami, On vector bundle valued harmonic forms and automorphic
forms on symmetric Riemannian manifolds, Ann. of Math. (2) 78 (1963), 365-416. MR 27 # 2997.
442
ARMAND BOREL
24.
, On certain cohomology groups attached to Hermitian symmetric spaces, Osaka J.
Math. 2 (1965), 1-35. MR 32 #1728.
25. K. Okamoto, On induced representations, Osaka J. Math. 4 (1967), 85-94. MR 37 #1519,
26. D. G. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94
(1971), 549-572, 573-602. MR 45 #7743.
27. M. S. Raghunathan, A note on quotients of real algebraic groups by arithmetic subgroups,
Invent. Math. 4 (1967/68). 318-335. MR 37 #5894.
28. J.-P. Serre, Cohomologie des groupes discrets, Prospects in Mathematics, Ann. of Math.
Studies, no. 70, Princeton Univ, Press, Princeton, N.J., 1971, pp. 77-169.
29. B. B. Venkov, On homologies of unit groups in division algebras, Trudy Mat. Inst. Steklov. 80
(1965), 66-89 = Proc. Steklov Inst. Math. 80 (1965), 73-100. MR 34 # 1379.
30. A, Weil, Adèles and algebraic groups, Notes by M. Demazure and T. Ono, Institute for
Advanced Study, Princeton, N.J., 1961.
INSTITUTE FOR ADVANCED STUDY
PRINCETON, NEW JERSEY 08540, U.S.A.