18
T. A. SPRINGER
ponent 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
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