18

T. A. SPRINGER

ponent G° is a reductive anisotropic fc-group. It is shown in the Bruhat-Tits theory

that if G is connected and simply connected simple &-group, there is a finite-dimen-

sional division algebra D with center k such that G(k) ^ SL(1, D).

If G is connected and reductive and is definable over o (which is always the case

if G is &-split, see [8, p.31]), then G(o) is a compact subgroup of G(k). There is a re-

duction map G(o) -* G(F), which is surjective.

4.8. Global fields. Now let A: be a global field. If v is a valuation of k, let kv be the

corresponding completion of k. If v is nonarchimedean, ov denotes the ring of

integers of kv. If S is a nonempty finite set of valuations of k, containing all the

archimedean ones, denote by os the ring of elements of k which are integral out-

side S. It is a Dedekind ring.

Let G be a connected reductive /c-group.

4.9.

LEMMA,

(i) There is an S such that G is definable over os;

(ii) G x

k

kv is quasi-split for almost all v.

(i) is easily established. Let C be the group of inner automorphisms of G; it is a

semisimple Agroup. We identify it with its group of fcs-rational points. There is

7 * e

Hl(k,

C) such that (7, twisted by a cocycle c from 7- (see 3.1), is quasi-split. This is

another way of saying that G is an inner form of a quasi-split group. Now c defines

a principal homogeneous space Cc of C over k, i.e., an algebraic variety over k, on

which C acts simply transitively, the action being defined over k (see [16, p. 1-58]).

We have y = 1 if and only if Cc has a ^-rational point. To prove (ii) it now suf-

fices to show that the image of 7 in Hl(kV9 C xkkv) is trivial for almost all v, or that

Cc has a /^-rational point for almost all v.

Let v be nonarchimedean such that C x

k

kv and Cc x

k

kv are definable over

o„, say C x

k

kv = C0 x

0v

kV9 Cc xkkv = CCt0 x

0v

kv. Assume furthermore that

the reduced group C0 x

0p

Fv, over the residue field Fv, is a connected ivgroup.

These conditions are satisfied for almost all v. By 4.4, it follows that CCt0 x

0v

Fv

has an irrational point. A version of Hensel's lemma then gives that CCt0 has an

Durational point, which shows that C has a Ayrational point. This implies (ii), as

we have seen.

4.10. Adelization. Let A be the adele ring of k. It is a Aalgebra, so the group of

v4-points G(A) of G is defined. Let G -* GLM be an embedding over k. Then g •-

(lf

£-1)

rnaps G(A) bijectively onto a closed subset of

An2+1

©

An2+1.

Endowed

with the induced topology, G(A) is a locally compact group, the adele group of G.

It has G(k) as a discrete subgroup. The topology on G(A) is independent of the

choice of the embedding G - » GL(«). An alternative way to define G(A) is as fol-

lows. Let S0 have the property of 4.9(i). For each finite set of valuations S = S0,

the group

G(AS) = n G(o.) x IT G(k9)

v&S

v-

S

is a locally compact group. If S a S' then G(AS) c G(AS,). G(A) can also be de-

fined as the limit group G(A) = inj limSzDSo G(AS) (this is independent of the choice

of S0).

For each v, we have an injection G(kv) - G(A).

EXAMPLES,

(a) G = GL(1). Then G(A) is the group of ideles (the units of A).

(b) k = g, G = SL(tf). One checks that