Download ON TAMAGAWA NUMBERS 1. Adele geometry Let X be an

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[И] L a n g S., Algebraic groups over finite fields, Amer. Journ. of Math.,
23, № 3 (1956), 555-563
[12] S a f a r e v i 5 I. R. (to be published).
[13] G r o t h e n d i e c k A., Le groupe de Brauer, II, Sem. Bourbaki, № 297
(1965).
ON TAMAGAWA NUMBERS
TAKASHI
ONO
1. Adele geometry
Let X be an algebraic variety defined over the field of rational
numbers Q. For each valuation v ( = oo or p) of Q, we get an analytic
variety Xv consisting of points of X rational over the completion QD.
If v —p, Xp contains a compact set Xz . Z p being the integers of Qp.
An element x = (xv) 6 \\ Xv is called an adele if xp £ Xz for almost
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p
all p, i. e. for all but a finite number of p. The set XA of all adeles
becomes a locally compact space and is called the adele space of X.
We identify XQ as a subset of XA by the diagonal imbedding. If X
is quasi-affine, XQ is discrete in XA.
The adele geometry is the study of the pair (XA, XQ), together
with the imbedding above. Therefore, one has to define all conceivable invariants of X in terms of the pair and study relations among
them or connections with other invariants of X. The Tamagawa
number x (X) is an example of such invariants which is, so far, definable only when X is a connected linear algebraic group. It is quite
desirable to find the true definition of т (X) for arbitrary X just as
the Euler number is defined not only for topological groups but for
any topological manifold. •
Having these in mind, we shall outline some known results on
algebraic groups and an interpretation of the Siegel's mean value
theorem as a statement about the Tamagawa number of a certain
homogeneous space.
2. Convergence property of a variety
For a variety X defined overQ, denote by X*p> the reduction of X
modulo p. For almost all p, X<p> is a variety defined over the finite
dimX
field F p = Zip and X[P Ф 0. We put \ip (X) = [X^]/p
, where
t*I denotes the number of elements in a finite set *. We shall say
that X is of type (C) if the product [JVP W> taken over almost all p,
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is absolutely convergent. When X = G, SL connected linear algebraic
group, we see that the following three properties for G are equivalent
each other: (i) G is of type (C), (ii) ô = 0, where ô is the character
module of G, (iii) ni (G) is finite. (We denote by щ (X) the /-th homo­
topy group of XQ.)
We shall call G special if it satisfies any one of the above three
conditions. By the decomposition of Levi-Chevalley, G is special if
and only if it is a semi-direct product of a unipotent group and a semisimple group. The condition (ii) can be replaced by (ii)': H° (X, 0£) =
= Gm, i.e. the non-existence of non-constant everywhere holomorphic never zero rational functions on X. Thus, all three conditions
(i), (ii)', (iii) make sense for any variety. It will be interesting to
study.their mutual relations for X. For example, the hypersurface
in affine (r + l)-space defined by F (X) = 2 tyX* — 6 = 0 , аг Ф 0,
4=0
b фО in Q, has all three properties, provided r > 3; actually Xc is
simply connected.
Suppose that X is non-singular. Let œ be a gauge form on X defined
over Q, i.e. an everywhere holomorphic never zero differential form
of highest degree defined over Q. Such a form exists if X = G or if X
is a hypersurface in an affine space. For each v, со induces a measure
CûD on Xv and we have, for almost all p, \ip (X) = \ CDP. If X is of
type (C), the formal product JJ a>v well-defines a measure on XA. If, in
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addition, X has the property (ii)', this measure on XA is the unique
one as is seen by the product formula in Q, and is written dXA,
3. Special groups
Let G be a special group defined over Q. By the argument above,
there is a unique measure dGA on GA. Since ô = 0, we have (G)Q = 0
and hence, by Borel—Harish-Chandra, the number % (G) = \ dGA
G
A/GQ
is well-defined. A. Weil has conjectured that
(W)
jt1(G) = 0^T(G) = l.
This has been proved for a large part of classical groups (Weil, Tamagawa), for some exceptional groups (Demazure, Mars) and for Chevalley groups (Langlands), but is not yet completely solved.
On the other hand, the author has determined т (G) modulo (W),.
the so-called relative theory, as an application of his determinationof the Tamagawa number of algebraic tori. Namely, since n± (G) is.
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finite, there is a universal covering group 8 of G which is also alge­
braic and defined over Q, unique up to isomorphisms over Q. Since
jti (G) can be identified with the kernel of the „covering map 8-+G,
we can put a g =: g (Q/Q)-module structure_on я4 (G), where g (Q/Q)
is the Galois group of the full extension Q/Q. The relative theory
tells that the ratio x (G)l% (G) depends only upon the g-module я4 (G).
More precisely, we have
(#)
% (G)lx(G) = [MG)ö]/[III (MG))b
1
where n/(G) = Hom (n^G), (Q)*), and III (*) = Ker (я (Q, *)->1
->• J J ^ (Q»» *))• Here, it should be noticed that the quantity on
the right hand side of (ф) makes sense for any X, at least when щ (X)
is finite. •
4. Special homogeneous spaces
Let (G, X) be a homogeneous space defined over Q. In particular,
(G,G) can be identified with G in an obvious way, and all definitions
for (G, X) will be consistent with respect to this identification. We
shall say that (G, X) is special if (i) XQ ф 0 and (ii) G, G6 (g £ XQ)
are special, where Gg is the isotropy group of £. When that is so, X
is quasi-affine (hence XQ is discrete in XA) and satisfies the three properties (i), (ii)', (iii) in § 2. Since G and Gg are unimodular, X has
a G-invariant gauge form and, by the uniqueness, X has the canonical
measure dXA which is independent of the choice of G and its action on
X that make X into a homogeneous space. We see that G^XQ is open
and closed in XA. For any / on GAXQ continuous with compact support,
compare the following two integrals:
J fdXA = (.)x(Qr* \
GAXq
%f(gt)dGA.
< V G Q ÊGXQ
If the ratio (*) is a constant independent of /, we shall call it the
Tamagawa number of (G, X) and write x (G, X). In case X = G,
.we have x (G, G) = x (G), this being nothing else than the Fubini
formula îOTGJGQ. By the mean value theorem we mean the statement
x (G, X) = 1 . For simplicity, assume that the universal covering
groups of G, G6 satisfy (W) and that G6 satisfies the Hasse principle.
Then, we can prove, via relative theory, that the mean value
theorem is the consequence of
rti(X) = jt 2 (X) = 0.