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0 such that, in the number-field case +00
F( 1N(x) 1)wA= c f F(t)dt/t , DA/Dk 0 f
resp., in the function-field case +00
f
DA/Dk
F( IN(x) 1)wA= c
whenever F is a function on
F(qV) ,
E
v=-oo
8+ (resp. on the group
{qVIVE~})
such
that the integral (resp. the sum) in the right-hand side converges absolutely. (N.B. In the function-field case,
q denotes the number of ele-
ments of the field of constants of k). Let first F be a continuous function with compact support on
the multiplicative group 8+ (resp. on the group {qv}); then lemma 3.1.1 shows that F(IN(x)l) has compact support in DA/D k, so that the integral in the left-hand side is convergent. For any to~ 8+ (resp. for any t o =qv) there is
XoE
DA* such that
IN(x)1 =t; repla0 0
c1ng then, in the left-hand side, x by xox, we see that it does not change if F(t) is replaced by F(tot); in other words, as a function Of F. it is translation-invariant in the group 8+ (resp. {qv}). This proves the lemma, for F continuous with compact support; the general
el •• follows from this in the usual manner. (For the value of c, cf. Th.orem 3. 1. 1. ) Now denote by Tr the reduced trace in Dk over k, which we
.xt.nd in the usual manner to a linear function on D, and also to a liMn mapping of DA = Dk®k Ak into Ak; Tr(xy)
is then a non-degenerate
- 34 -
bilinear form on OxO. If X is the character of Ak defined in Theorem 2.1.1, we put XO(x) =X(Trx)
x~
for
0A; then Xo has the proper-
ties corresponding to those stated for X in Theorem 2.1.1: XO(xy) determines an isomorphism of 0A onto its dual group, and Ok is selforthogonal for this isomorphism. As a consequence, if we define the Fourier transform
~(y)
~(x)
of a function
on 0A by the formula
we have the inversion formula
and the Poisson summation formula L
~(Il)
under suitable conditions for are valid if both
~
such that the series
and
~
=
'1'([3)
L
f3£ Ok
IltO k ~
and
~.
For instance, these formulas
are continuous, absolutely integrable, and
La~(X+Il), L[3~(y+[3)
convergent; when that is so, we say that
are absolutely and uniformly ~,
~
are "of Poisson type".
If, in the definition for ~,we substitute xa, a- 1y for x, y, where aEO A, we see that the Fourier transform of ~(xa)
is
IN(a)l- n'l'(a- 1y);
similarly (and in view of the fact that XO(axa -1) = XO(x), by the definition of XO) the Fourier transform of ~(ax) Now we say that ~(ax)
and
~(xa)
~
is
IN(a)l- n'l'(ya- 1).
is of standard type if, for every
a~
0A'
are of Poisson type and if the following conditions
are satisfied : (a)
There is an S (i.e. a finite set of valuations of k, including
all the infinite places) such that, for x = (xv)E-. 0A' Xs = (x)ve.S :
- 35 -
where ¢
p
is the characteristic function of 00 ' and -p
nuous and absolutely integrable in
Os
~S
is conti-
=TIvES Ok
. If this condition is v satisfied for some S, it is clearly also satisfied for every S'=>S.
Also, it implies that the Fourier transform
~
of
~
satisfies a simi-
lar condition (but possibly with another S). (b)
The integral
and the similar integral for
~,
converge absolutely and uniformly for
s ~ O. One special method for constructing functions of standard type is as follows. In the function-theoretic case, take any set S as in (i), and take for
~S
the characteristic function of any compact open
subset of OS' or a finite linear combination of such functions. In the number-theoretic case, let So so that Os
o
= Ok Ok
be the set of the infinite places of
8 ; on Os
k,
8,
considered as a vector-space over 0'
let P be a polynomial function and
F a positive-definite quadratic -F(x ) x C:O S ' take ~ (x ) = P(x)e 0; on the other hand, let o 0 0 0 0
,form; for
S· be any finite set of valuations of k, disjoint from So' and let ••
be the characteristic function of a compact open subset of OS"
I finite 1inear combination of such functions; take
DS • DS
o
x
DS" and, for
th.n define
~
xo c: Os ' x' E: OS', take
or
S = So US', so that
~S(xo,x') = ~o(xo)~'
(x');
0
as in (i). It is easily seen that, in both cases,
~
is
Of standard type, and that its Fourier transform is another function of
the same nature. Definition. Let ¢ be a function of standard type on
0A; the
- 36 -
function (2)
will be called the zeta-function of D with respect to
~.
This definition will be justified by proving, firstly, by a multiplicative calculation, that the integral for Z~(s)
is absolutely
Re(s)~n+£
convergent for Re(s»n (and uniformly so for
for every
£ >0), and, secondly, by an additive calculation, that it can be continued analytically in the whole s-plane. For the multiplicative calculation, take S as in (a) and put, for every S'.:::>S:
= TT Dk , Dt S' ) =DS' x TT D*
DS'
VES'
V
t
PfS'
Qp
Then, by definition of the adele-space, we have
where the limit is taken over the filter of the sets S', ordered by inclusion. Now call
Z~(s)
the integral (1); this is the same as the same
integral taken over DS' since
l1 vES IN(x) Iv
is 0 on DS - DS. Also,
put, for p¢ S
and denote by Z~(s)
where, as before,
the same integral taken over D* ; then 2p
A
p
=1-N(p)-1
for a p-adic valuation, and AV = 1 for
- 37 -
an infinite place. Now, on 0*, we have IN(x p) Ip = 1 (provided S has 2p been taken large enough) and ¢p(x p) = 1, so that Z~(s) is the wI-measure of 0*; by the definition of convergence factors, this imp 2p plies that
TTpZ~(s)
is absolutely convergent. Therefore
As we have seen, we can identify 0
with Mn(2p) for p¢ S provided 2p S has been taken large enough. Write Mn (2 p)* for the set of the ma-
trices in Mn(o) -p with a non-zero determinant, and Un, p for the set of the invertible matrices in Mn(2 p)' i.e. those whose determinant is a unit in 0; -p we have, for
pES and q=N(p) :
Zp (s) = J ) det X 1 ~ (1-q -1) -1 1 det X 1 ~n (dX) p Mn(2p) For a given p, call
TI a prime element in 2p
(this means that TI2p
is the maximal prime ideal in 2p). It is well-known that Mn(2p)
is
Un,p A when one takes for A the follo-
the disjoint union of the sets wing matrices :
d1 71
d2 71,
A=
a ij
,
0
, 71
dn
where Cd1 , ••• ,d n) run through all n-tuples of integers
~ 0,
and, for
tach choice of the d., each a·· runs through a complete set of repred.
1
I.ntatives of 2p mod
71
lJ
J. This gives
( 1-q -1 ) Zp (s) = ~ 1 det A 1 ~ U J AI det X 1 ~n (dX) p n,p
- 38 -
As the integrand in the right-hand side is (by construction) invariant under translations in Mn(k p)' the integral is independent of A, taking A= In Un,p v
(the unit-matrix), we see that its value is the measure of _n 2 for the measure (dX)p' in the space Mn(k p); this is q v if
is the number of matrices of non-zero determinant in the ring
Mn(F q ), where
Fq
is the finite field with
q elements; this gives
U f (dX)p = (l_q-n)(l_q-n+l) ... (I_q-l) n,p On the other hand, for given values of d1, ... ,d n , there are
matrices
A. Therefore: Z (s)
=
(1_q-n) ... (1_q-2)
Z
,
Z.(i-l-s)d. q' ,
(d.)
p
From the elementary theory of the zeta-function for fact that the product' sk(s) =lTp( l-q -sf' s real and> 1, and that
(s-l)sk(s)
npctS Zp(s)
and is -p(s-n)-l, with
P a constant, for
converges absolutely for
asserted, the integral which defines
where
is absolutely convergent for
tends to a finite limit
s->-1; th,: implies that
for
k, we borrow the
Z¢(s)
Re(s) > n. Also, we have
Pk depends only upon the field
k.
Pk for Re(s) > n
s ->- n. This proves that, as is absolutely convergent
- 39 -
For the additive calculation, introduce on A(t)
defined by A(t)=l
6+ the function
for On, we conclude that z!(s)
Z!, converges absolutely
is an entire function of s. On
the other hand, we have, for Re( s) >n
(Dl/D~
is the space of right co sets
in DA). By Poisson
xD k of Dk
lummation, we have 1:
aeDk
Whlre
'l'
(xa) = IN(x)
rn 1:
'l'(Bx- 1) , BeD k
is as before the Fourier transform of
;
hence
- 40 -
this still being absolutely convergent for gral defining n- s
for
Re(s)
>n.
Now, in the inte-
Z+(s), which converges absolutely for all
sand
s, substitute
x- 1 for X; as the latter substitution does not
affect the Haar measure, and as
f+(X- 1 ) =fJx), we get
, Z'¥(n-s) = ffJx)IN(x)l s - n '¥(x- 1)wA
D*
+
A
fJx)IN(xl 1s - n( L:
f
=
DA/Dk
SEDk
'¥(sx- 1 )\wA
J
Combining our two last formulas, we get Z'(x')d'x' = J (
G'
H'
L:
4>'(x'~'))d'x"
(z'=z6).
f4>"(z)d z = J ( L: 4>"(u;))d z' 9
y ~6
9
(x"=x'll'),
~'E:ll'
9
Combining these, and using the fact that 9 and 6 are in the center, we get (1)
J(xzO)d z' H'
Y ~tll
(z' = z6, u' = xgll) .
9
On the other hand, we have (2)
J (xO )du G
H
(u=xll).
~Ell
The comparison of (1) and (2) concludes the proof, since there are "sufficiently many" functions
L:~Ell 4>(x~)
on H.
Now, in Lemma 3.2.1, we replace G, ll, 9 by DA, Dk, ZA' respectively, where Z is, as before, the center of the algebra variety
D. As in 3.1, we write G for the algebraic group D*/Z*, i.e. for the projective group of D; Z* may be identified with Gm; as it is well known that every fibering by Gm has local cross-sections, we can apply Theorem 2.4.2 to D* and Z*, and therefore identify DA/Z A with GA; also, if we identify Z* with Gm, ZA gets identified with the idele-group of
(Gm)A=I k,
k. On DA and DA/D k, we take the measure denoted
above by w'A' The proof in 3.1, applied to the case n = 1, shows that the set of convergence factors
(Ap)
which was used to define wA is
a1so a set of convergence factors for Z* = Gm; we denote by
z
W
the
Tamagawa measure determi ned by thi s set on ZA = I k' and also on
ZA/Zk = Ik/k*. Then, by Theorem 2.4.3,
(1)
is a set of convergence fac-
tors for G= 0* IZ*; we denote by wG the Tamagawa measure on GA, and
- 43 the corresponding measure on
GA/G k. In view of Theorem 2.4.3, we can
now apply to this situation our Lemma 3.2.1, and get
where
F is any function such that the left-hand side converges absolu-
tely. The left-hand side can be expressed by means of Theorem 3.1.1 (iii); on the other hand, the second integral in the right-hand side can be written as
which can be expressed by Theorem 3.1.1 (iii) applied to Z instead of
o
and is thus seen to have the value P
+00
+00
Pkf F(IN(x)lt n)dt/t=.J5 f F(t)dt/t a n 0
in the number-field case, and +00
Pk log q
L:
F( IN(x) Iq\!n)
\)=-00
in the function-field case. Comparing both results, we see at once, in the former case, that
me
has the value n; we get the saA k conclusion in the latter case by taking, for instance, F such that
'(q\l) = 1 for
T(G) =fG /G wG
a ~ \! ~ n-1,
and = 0 otherwise. Thus:
Theorem 3.2.1. The Tamagawa number of the projective group of
I division
algebra of dimension
n2 over its center is
n.
3.3 Isogenies. We recall that an isogeny is a homomorphism of an algebraic
,roup onto another of the same dimension; two groups G, G' are called
- 44 isogenous if G" can be found so that there are isogenies of G" onto G and onto G'. In this section, we consider Tamagawa numbers of groups isogenous to projective groups of simple algebras, and products of such groups; this, combined with Theorem 3.2.1, will give for instance the Tamagawa number of the special linear group of a division algebra. Lemma 3.3.1. If two groups G, G'
are isogenous over k, every
set of convergence factors for G is a set of convergence factors for G' •
Assume that there is an isogeny f of G onto G' by means of representations of G, G'
over k;
into special linear groups, we
can consider them as affine varieties; then, if x' = f(x), the coordinates of x'
can be written as polynomials in those of x. As in 2.2, we
see that G, G'
and f can be reduced modulo
p for almost all
p.
Our lemma is now an immediate consequence of Theorem 2.2.5 and of Lang's theorem, according to which two isogenous groups over a finite field have the same number of rational points (Am. J. of Math. 78 : see last five lines of p. 561). Any simple algebra
R can be written as Mm(D), where D, as
in 3.1, is a division algebra. The same calculation as in 3.1 shows that the
Ap =1 - N(p) -1
is a set of convergence factors for R*; as it is
also a set of convergence factors for
Z*, where Z is the center of
R, Theorem 2.4.3 shows that (1) is such a set for R*/Z*, hence also for every group isogenous to R*/Z* (Ap)
(in particular, for
is such a set for every group isogenous to
R(1)) and that
(R*/Z*)
x
Gm. In what
follows, we shall use these facts freely. Tamagawa measures in the strict sense (derived from the set (1) of convergence factors) will be denoted by w, wA' dx, etc.; by w', wA' d'x, etc., we denote Tamagawa measures derived from the set of convergence factors (1_N(p)-1); for instance, on
I k, we use the Tamagawa measw'e (dt/t)'.
- 45 -
As in 3.1, let
Dk
n2
be a division algebra of dimension
over its center Zk = k; D being the algebra variety defined by the center
Z, take
R= Mm(D); call
center. Take any divisor
N the reduced norm in
v of mn (1
D, with
R over its
Sv Smn). Let r be the group
r ={(x, v) c R* x GmIvv =N(x)} , i.e. the algebraic subgroup of connected and defined over
R* x Gm determined by
VV =
N(x); it is
k. The connected component of the identity
in its center is
and is obviously isomorphic to Put
Gm; here
1R
is the unit-element in
G= fir 0; this is an algebraic group, defined over
is isomorphic to
R*
and
tive group of R; for momorphism of r identified with R(1) +G+R*/Z*
G can be identified with
v = mn, the mapping
onto
R( 1)
k; for
R.
v = 1, r
R*/Z*, the projec-
(x,v) +v- 1x determines a ho-
with the kernel
fa, so that
G can be
R(l). For any v, there are obvious isogenies such that the composite ;sogeny R(1) +R*/Z*
is the ca-
nonical one. Theorem 3.3.1. We have
'r(G) = mn/v.
Consider the homomorphism ¢ of
r
onto
Gm given by
.(x,v) =v; its kernel is
and is isomorphic to R(1); by means of
1m'
as usual, we extend
¢
¢, we can identify
to a homomorphism of r A into
r/r'
with
(Gm)A = I k .
Lemma 3.3.2 ¢(rA)/¢(r k) = Ik/k*. It is known (Eichler, Math. Zeitschr. 1938) that an element
A
- 46 -
of k*
is the norm of an element of Rk
if and only if it is the norm
of an element of Rk
for every v; the latter condition is equivalent v to saying that A must be the norm of an element of RA. Now, for
ack*, we have aE¢(f A) if and only if aV of RA; then, by Eichler's theorem, Rk. This proves that
aV
is the norm of an element
is the norm of an element of
k*n¢(f A) =¢(fkl. Now we show that
In fact, if x = (xv) E.. I k, it is known that xp ment of Rk Rk
for every p, and that Xv
Ik = k*.¢(f A).
is the norm of an ele-
is the norm of an element of
p
whenever v is a complex place (i .e. kv
=~),
and al so whenever v
v
is a rea 1 place (i. e. kv = ~) and Xv >o. Moreover, for almost all p, Ro = Mmn(Qp)' so that the image of f by ¢ is (Gm) = Up (the -p Qp Qp unit-group of 0). Algebraically, our conclusion follows at once from -p these facts; the same holds topologically (so that we may identify ¢(fA)/¢(f k) with Ik/k*) As
in view of the final remark of Chapter II.
(x ,v) .... x is an isogeny of f
onto R*, (1-N(p)-1)
set of convergence factors for f; let d' (x,v)
is a
be the corresponding
measure. We shall discuss the number-field case; the function-field case can be treated quite similarly. We compute in two ways the integral J
F(\v\)d'(x,v)
fA/fk where F is an arbitrary function (say, continuous with compact supporr) on the group
~+.
We fi rst use the decompos iti on G= fir 0; as r 0
"
Gm,
it has cross-sections in r, so that we can apply Theorem 2.4.3. Then, by Lemma 3.2.1, we have J F(\v\)d'(x,v) = rA/r k
the second integral in the right-hand side can be computed by Theorem
- 47 3.1.1 (iii) applied to
Gm, which shows that it is independent of v
(this would have to be modified in the function-field case, just as in the last part of the proof of Theorem 3.2.1), and gives p v f F(lvl)d'(x,v) = '[(G) ;n JF(t)dt/t rA/r k 0 00
On the other hand, applying Theorem 2.4.4 and Lemma 3.3.2 to the decomposition
Gm= r/r', we get:
where the right-hand side can again be computed as above. This shows that, if one of the numbers
'[(G), '[(r') is finite, the other is so, and
that '[(r') = v'[(G)/mn. Take m=v=l; then and
r' =D(l), G is the projective group of D,
'[(G) = n by Theorem 3.2.1;' therefore '[(D( 1)) = 1. Take m= 1, and
take for v any divisor of n; then
r' =D(1), T(r')=l; this gives
'd6) = n/v, and proves Theorem 3.3.1 in the case
n = 1.
In the general case, we have '[(G) =mnc(r')/v, with
r' ~R(1).
Thus, in order to complete the proof of Theorem 3.3.1, it only remains to show that '[(R(1))=1
for
R=Mm(D); as we know that this is so for
m. 1, we shall proceed by induction on m. 3.4. End of proof of Theorem 3.3.1 : central simple algebras. We change our notations slightly. From now on,
D will be as
before; we write Rm=Mm(D). I~e denote by Dm the space of (m,l)IIItrices (i.e., column-vectors of order m) over D, and let Rm act on
,om
by
(X,x)"'Xx
for
X£Rm, xEDm. Put
- 48 -
e=
(we write
(1)
1 for the unit-element of D). Call
is generic over kin Dm if X
der the action of is so in
Rm, H is a Zariski-open subset of Dm over k. More preci-
sely, if K is any field containing lows. As
k, HK can be determined as fol-
DK = Dk0K is a simple central algebra over K, there is an
isomorphism
p
division algebra
of DK onto a matrix algebra Mr(D') D'
over a central
over K; then HK consists of the column-vectors
x = (x i )1 < i <m' XiE.D K, such that the
p(x)
over D'
H the orbit of e un-
(mr,r)-matrix
=
has the rank r. From now on, we assume that m~ 2, and we put G = R~ 1) ,
G' = R(1). it is easily seen that
H is also the orbit of e under G. m-1 ' Call g the subgroup of G leaving e fixed; it consists of the matr~ ces 1 X= ( 0
(tu
tu) X'
( u E Om-1, X' E. G' )
is the transpose of u). If x=(x i )1'f(O), we see that T(G) = 1. This completes the proof of Theorem 3.3.1. 3.5. The symplectic group. Using the same method as in 3.4, we prove Theorem 3.5.1. As usual,
Sp(2n)
T(Sp(2n)) = 1. is the symplectic group in 2n variables,
i.e. the subgroup of GL(2n) which leaves invariant the exterior form
- 53 -
We have
Sp(2) = SL(2), so that
T(Sp(2)) = 1 is a special case of Theo-
rem 3.3.1. Now we proceed by induction on G=Sp(2n), G
=Sp(2n-2); call column-vector e = (1,0, ••. ,0) I
n~2;
n. Take
g the subgroup of
put
G leaving the
invariant. An easy calculation shows that
g consists of the matrices
x
u XIE: GI ,
where and of
u is a column-vector of order
g, consisting of the matrices
more precisely,
g is the
moreover, the subgroup
duct of
GI . The subgroup
is the alternating matrix invariant under
JI
which
2n - 2, x is arbitrary,
XI
g"
X for which
product of
~emidirect
of
g",. Ga and of
induction assumption
XI = 12n - 2 , is normal;
gl
glgl,. G
and of
I ;
gl, consisting of the matrices
= 12n - 2 and u = 0, is normal, and gl
gl
X for
is the semidirect pro-
ql • Ig" ,. (G a )2n-2. From these facts and from the
T(G I ) = 1, one concludes, by applying twice Theorem
2.4.3, that (1) is a set of convergence factors, first for
gl, and then
for g, that T(gl) = 1, and that T(g) = 1. Now the orbit of e under G
15 H= S2n - {O}, where S2n is the affine space of all column-vectors Of order 2n; just as in 3.4, we see that we can identify H with :8/9. HA wi th
GAl gA' etc.; a 1so,
HQ
p
cons is ts of the vectors in
02n -p
which are not :: 0 mod p ; an easy calculation shows then that (1) is a
,.et
of convergence factors for
H as well as for
S2n, so that we can
apply Lemma 3.4.1; also, by Theorem 2.4.3, (1) is a set of convergence
factors for G. Now we can apply Lemma 2.4.2 to GA, gA' Gk , gk; this
Itves
- 54 -
f f(x)dx
f (l: f (XC;) ) dX GA/G k C;C:H k
=
HA
(A k)2n; also,
In the left-hand side, we can replace HA by
Hk
is the
k2n - {O}. Proceeding now exactly as in 3.4, we get T(G) = 1.
same as
3.6. Isogenies for products of linear groups. The method of 3.3 can be applied to a more general situation. Let R be any absolutely semisimple algebra-variety over k; this is the same as to say that Rk
is absolutely semisimple, or also that R, over
the universal domain, is isomorphic to a direct sum of matrix-algebras. We can write
1.s. i .s. r, are the simple com-
R= ® Ri' where the Ri' for
ponents of R; for each
i, we have Ri =Mm. (D i ), where Di
is such
1
that
(D i \
is a division-algebra. If Zi
is the center of Di' which
we identify in an obvious manner with the center of Ri' (Zi)k a finite, separably algebraic extension of k, and
=
ki
is
(Di)k may be consi-
dered as a central division algebra over ki' which can be written as (D~)k' 1
where
i
Dl~
is an algebra-variety over
the notations of 1.3,
Di
=
ki ; then we have, with
Rk./k(Djl, hence Ri
Rk./k(Ri)
=
1
R~ = M (D ~ ); a 1so, if 1 mi 1
Zi
=
Zi
with
1
is the center of Di
and Ri, we have
Rk./k(Zjl; and the reduced norm Ni' taken in Ri
over its center,
1
determines (by applying to its graph the operation ping Ni
of Ri
into Zi. We write
R~1)
for the
Rk ./ k) a norm map-
sP~cial
linear group
of Ri' i. e. the algebra i c subgroup of Ri determi ned by Ni (x) = 1; Rk ./ k(Ri(1)), and we have, by Theorems 1.3.2, 1.3.1
this is the same as
1
(R~ 1) ) k
(R ~ ( 1) )Ak.
=
(R i ( 1) )Ak
1
which, in view of Theorem 2.3.2, implies T(R~1)) 1 of the operation
=
1. By the definition
Rk ./ k, the algebraic group R~1), over the universal 1
- 55 -
d.
domain, is isomorphic to
(SL(min i ))
1
where di = [k i : kJ
and n2i is
the dimension of Di. Write N for the mapping
of R into its center Z = . (i) ) v
is a set of convergence factors for
Rki/k (Ri*); such a set can be chosen at once, by Theorem 2.3.2 and
- 58 -
the results of 3.3, by putting
Av(i) = 1 whenever v is an infinite
place of k, and otherwise :
where the product is extended to all the prime divisors ki' and the norm N(P)
P of p in
is the absolute norm (equal to N(p)f if P is
of relative degree f over pl. For the same reasons, this same set of convergence factors can also be used for Z*, hence also for r o ' and (by Lemma 3.3.1) for the torus T. On the other hand, the same argument shows that (1) is a set of convergence factors for R(l), hence for r', and also for G which is isogenous to R(l). vIe denote by d"(x,t) Tamagawa measure for rA' with the set of convergence factors
the
(A V)' and
use similar notations for Z* and T. Now we compute in two different ways the integral
here
I I
denotes as usual the idele-module, and F is an arbitrary
function (say, continuous with compact support) on the group
(~+)r; we
do this by using the two decompositions G=r/r o and T=r;r'; this will be carried out in the number-field case (the function-field case can be treated similarly, mutatis mutandis). The first decomposition gives (1)
while the second one gives, since T(r') = 1 : (2)
- 59 -
For each
i, we can identify
(Z·PA = (Rk./k(Gm))A with
(Gm)A
1
therefore ZA
ki
= I k.; 1
is the same as TTil k ., while Zk is the same as 1
TIiki; and the idele-module idele-module
Iz. I·
1 1
Ivi(zi) I, taken in
taken in
I k , is the same as the
I k . From these facts, and from Theorem i
3.1.1 (iii), we conclude that we have
whenever f is such that the
right-hand side is absolutely convergent.
This gives for (1) the value:
As to (2), we apply Lemma 3.6.1, together with the following remark: for every tE:T A, there is l~i~r
(x' ,t') E:fA such that
IXi(t-1t') 1= 1 for
(in fact, it is easily seen that we can take x' =z, t'
=~(z),
for a suitable z £ ZA); therefore, we can choose, as representatives of the
2i
for all
cosets of ¢(f AlT k in TA, elements t
such that
IXi (t) I = i
i. From this, one concludes that the integral in the right-hand
side of (2), taken over everyone of the
2i
cosets of ¢(fA)/¢(f k) in
TA/T k, has always the same value. Therefore: (4)
The comparison between (3) and (4) shows that (apart from the determina-
t10n of the index 2i , which is effected by the result of Serre and Tat~ ~.
computation of T(G)
~l.m concernin~!J~
has been reduced to a purely commutative pro-
torus T. viz. the computation of the integral in (4).
- 60 -
This problem has been studied by Ono (Ann. of Math. 1961) but is not yet completely solved. Before making some applications of the results obtained above to the orthogonal groups, we insert a few general remarks about toruses. Let T be any torus over k; by Theorem 2.2.2, its characters (i.e., its representations into Gm over the universal domain) make up a finitely generated free abelian group (free, because T is here assumed to be algebraically connected), on which the Galois group, of k/k (k = algebraic closure of k) operates in an obvious manner (actually, every character of T is defined over a separably algebraic extension of k). This group, written additively and considered as a representation-module for jt
,
is denoted by T and is known as the dual module of
T; Tate has shown that there is a duality of the usual type between such modules and toruses over k. We write Tk for the group of the elements of T which are invariant under
~
(these correspond to the characters
of T which are defined over k). Let tation of
0;
be the trace of the represen-
~
(with coefficients in Q) given by the operation of
on the vector-space TQ0g over the "character" pole of order r
~
of if r
!I
Q;
ff
according to Artin,there belongs to
an L-function
L(s,~),
which has at s = 1 a
is the number of generators for Tk; this
L-
function is given by an infinite product L(s,~)
=
np Lp(S,~)
taken over all the p-adic valuations of k. Now, combining Theorems 2.32 and 2.4.3 with a theorem of Artin on rational representations of finite groups, we see that, by putting "v=1 ptS
(as usual,
for
VE:.S,Ap=Lp(l,~)
for
S is a finite set of valuations of k containing all
the infinite places), we define a set of convergence factors for T. If
- 61 -
at is the Tamagawa measure constructed by means of that set, we have a formula
in the number-field case (the integral in the right-hand side is to be replaced by a series, in the usual manner. in the function-field case); here X1""'X r are generators for the group of characters of T, defined over k; moreover, if we put
the constant r(T) = c/p is independent of the choice of S and of the generators Xi
and is invariantly attached to the torus T; it has ob-
vious "functorial" properties such as r(T1 xT 2 ) = r(T 1)r(T 2), and r(Rk'/kT')=r(T')
if T'
is a torus over k'.
If now notations are again as in (3) and (4), our results (taking into account the theorem of Serre and Tate) give T(G)= r(T)ldet(aij)l. This raises the question whether, at least for toruses isogenous to Z*, the number r(T)
is always an integer.
3.7. Application to some orthogonal and hermitian groups. In view of the well-known "canonical isomorphisms" between classical groups, the Tamagawa numbers for the orthogonal groups in 3 .nd 4 variables and for the hermitian groups in 2 variables can be cal-
culated by means of the above results; this will provide the starting
, point for the consideration of the orthogonal groups, by induction on the number of variables, in Chapter IV. We avoid complications by excludino once for all the case of characteristic 2 (there is, however, no ,.•• on for thinking that our main results do not remain true even in
- 62 -
that case). A quadratic form of index
F, with coefficients in
0 if it "does not represent"
iution in
k, other than
0, i.e. if
k, is said to be F(x) = 0 has no so-
O.
(a) Orthogonal group in 3 variables : Let form in 3 variables, and
F be a quadratic
G the "speciaj" orthogonal group of
("special" = determinant 1) ~ then
F
G is isomorphic to the projective
group of a simple central algebra of dimension 4. viz. a quaternion algebra if
F is of index 0, and the matrix algebra
M2
otherwise; bv
Theorem 3.2.1 in the former case, and by Theorem 3.3.1 in the latter case, we have
T(G) = 2.
(b) Hermitian group in 2 variables: let extension of
k; take
Z'
over
k', such that
k'
be a quadratic
Zk' = k'; take
R' =M 2 (Z'), Z=Rk'/k(Z'), R=R k '/k(R')=M 2 (Z), so that z ->- Z of
non-tri vi a 1 automorphi sm
k'
over
Rk =(M 2 \,. The
k can be extended in an
obvious manner to an automorphism of the algebra variety k, whi ch we also denote by
over
X->- X of Z*
R, defined over
defi ned by
written as
k. We can identify
z = z: the norm mappi ng of
form
Rk = (M 2 \, such that tF(x) = x,S'x on the space
is said to be of index other than
Z*
Gm with the subgrouD of into
Gm can then be U of
S be an invertible hermitian matrix over
element of
0 if
Z, defined
z ->- Z, and then to an automorphi sm
z ->- zz. and its kernel is the subgroup
zz=1. Now let
Z*
defi ned by
k'. i.e. an
ts = S~ this determines the hermitian 2 x1 Z of vectors x = ( ) over Z; F
x2
F(x) = 0 has no solution in
Z2k -- k,2 ,
O. The hermitian (or "unitary") group attached to
F(x), is the subgroup
G of
R*
D' = Z',
S, or to
given by :
G= {XE R*I t X' S' X= s, N(X) = 1} where
N(X) = det(X)
is the reduced norm taken in
It is known that
R over
Z.
G is isomorphic to the special 1i near group
- 63 R;l)
of a simple central algebra
Quaternion algebra if otherwise. Therefore
Rl
F is of index
of dimension 4 over
0, and the matrix algebra
M2
T(G) = ,.
(c) Orthogonal group in 4 variables : let form over
k. viz. a
k, in 4 variables,
6
F be a quadratic
its discriminant,
G the "special"
l:
orthogona 1 group for
F. If 6 2 E. k, there is an algebra
sion 4 (a quaternion algebra if algebra
M2 ) such that
F is of index
used in 3.6, this can be written as R=R 1(±)R 2 , hence for
of dimen-
0, otherwise the matrix
G= (R;l) x R;l))/y, where
order 2 consisting of the elements
R,
is the subgroup of
y
(1,1), (-1,-1). With the notations G= r;r 0
when we take
Rl = R2 ,
Z*= GmxGm, N(z)=z2, T= GmxGm, lJ(z)=(z,z2,z11z2)
Z= (zl,z2)€Z*' and
v(t)= (tltzl,t1t2)
for
t=(t 1,t 2 )tST. The
integral in (4) can be calculated by Theorem 3.1.1 (iii) and has the value
also, one finds at once that
1.
Now assume that Then
2i = 1, Idet (a ij ) 1= 2. This gives
k' = k(6 2 )
G is isogenous to
is a quadratic extension of
Rk '/k(R,(1)), where
R'
T(G) = 2. k.
is a central simple
algebra of dimension 4 over k' (again a quaternion algebra or M2
ding as
F is of index
(R,(l)XR,(l))/y, with
0 or not); over y
as above. Let
acco~
k', it becomes isomorphic to Z', Z, U and the mapping
Z + Z be defi ned as in (b). Then we can write
G, with the nota t ions of
3.6, as G=r;r o for r=1, k1 =k', Ri=R', R1 =R, N(z)=i, T=GmxU, II(Z)= (zz,z-l z) for
z€.Z*, v(t)= (t1tz1,t1t2)
t,€Gm, t2€U. We have now a 11
'or
=
for
t= (t 1,t 2 ),
1, 2i = 1. Now we have to calculate (4)
T" Gmx U. As we take our Tamagawa measure for
Gm by means of the
',ctors Ap = 1 - N(p)-1. and for Z* and T by means of the factors
- 64 A' =
P
where the product is taken over the prime divisors
p'
of p in
k',
we have to take the Tamagawa measure on U= T/Gm by means of the factors
where X is the character associated with the quadratic extension
k'
of k. Applying now Theorem 2.4.3 and Theorem 2.4.4 (the latter, in the modified form explained in the Remark following it) to the groups Z*. U, Gm= Z* /U and to the norm mappi ng z -+ zz of Z* onto Gm, we get (5)
where HA, Hk are the images of ZA' Zk under the norm mapping, T'(U) = fU /U dAu, and
dAz, dAY' dAu are the Tamagawa measures for A k Z*, Gm, U constructed by means of the sets Ap' Ap ' A~ defined above. In the right-hand side of (5), ZA' Zk may be identified with
with
k'*; as
II
is the idele-module taken in
same as the idele-module taken in
Ik,lzzl
I k , and
is then the
I k,; therefore the right-hand side of
(5), computed by Theorem 3.1.1 (iii), is 00
Pk' f F(t)dt/t
o
On the other hand, by class-field theory,
HA/H k, in the left-hand side,
is nothing else than the open subgroup of
Ik/k* of index 2 determined
by X(Y)
=
1, where X is the character of
dratic extension
Ik/k* belonging to the qua-
k'/k. Therefore the integral in the left-hand side has
- 65 -
the value 1
iPk f F(t}dt/t
o
This gives ,'(U) = 2Pk'/Pk ,'(U)
=
(which can also be written as
2L(1,X». As the integral in (4) has the value 00
,'(U)
f
F( Ixl )(dx/x)'
Ik/k* we get, as before,
,(G)
=
=
,'(U)Pk f F(t)dt/t , 0
2
Theorem 3.7.1. The Tamagawa number of all special orthogonal groups in 3 and 4 variables has the value 2. 3.8. The zeta-function of a central simple algebra. We have already twice made use of results in class-field theory (in the latter part of 3.8, and implicitlY by using Eichler's norm theorem in the proof of Lemmas 3.3.2 and 3.6.1); we shall also use such results freely in our treatment of the classical groups in Chapter IV, both directly and by our use of Hasse's theorem on quadratic forms (whkh can be derived formally from the norm theorem for quaternion algebras).
It is known, on the other hand, that most of these results can be derived from Hasse's theorem according to which "a central simple algebra wh1ch splits locally everywhere splits globally". This will now be proved by a more precise calculation of "the" zeta-function of an algebra (1ndependently of our use of class-field theory in 3.3, 3.6, 3.8). It is therefore likely that, by following up this idea, our treatment could be rendered completely self-contained. The multiplicative calculation of Z¢(s)
for a division alge-
bra 1n 3.1 can be extended to any central simple algebra; this will be done (following Fujisaki) for a special choice of ¢.
- 66 -
Let R be an algebra-variety over k, with the center Z, such that Zk = k and that Rk is a simple algebra; let n2 be the dimension of R (equal to the dimension of Rk over k). Take a basis (u.) 2 of Rk over k. As we have observed before, there is S 11
-N(X)
Rk , /k (Gm)
maps the
determi ned by
ZZ = 1, and G is the kernel of that homomorphism. Now we put
F(x) = ti(Sx, with
x cOm; as the characteristic is
not 2, S is uniquely determined by the values of
F on
m
Ok' except in
- 75 -
the case S1, when F = 0;
F is called "quadratic" in the case 01,
"hermitian" in cases L2, S2, "anti hermitian" in the case 02; it is a scalar polynomial function in cases L2(a), 01, S2; in the case 02, it takes its values in the (3-dimensional) subspace D of odd elements of
x=
the quatern i on algebra D (the elements such that
-
x); in the case
L2(b), it takes its values in the n2-dimensional space D+ of even elements of D (the elements of D such that
x= x).
Writing T for anyone of the symbols L2(a), L2(b), 01, 02, S1, S2, we say that the group G defined by (1)
is of type Tm' With
this notation, we have the lemma Lemma 4.1.1. Let G be the group of type Tm defined by (1); ~
~
be a vector in ~
G leaving F(~)
f 0,
F(~) = 0, F(~) =
G"
D~, other than
fixed. Then: (a)
0; let g be the subgroup of
2i. m= 1, g = {e}; (b) if m> 2 and
g is isomorphic to a group of type Tm- 1; (c) if m= 2 and g
.!E. {e} or isomorphic to a group (Ga)r; (d) if
0, g is the semidirect product of a group g'
of type Tm- 2 , where g'
m~ 3 and
and of a group
is either isomorphic to a group
(Ga)r or
to the semidirect product of two groups of that type. For m= 1, we must have (b), space
F(~)
f 0;
and (a) is trivial. in case
g is isomorphic to the group of type T, acting on the vector-
t~Sy = 0 and 1eavi ng i nvari ant the form induced on it by F. In
the cases (c), (d), one can, by a suitable change of coordinates over transform S and
~
into matrices
o o Then 9 consists of the matrices
:
S"
),
~ (~) =
0
~
- 76 -
with
x:!: x + tus"u = 0, v = :!: tus"X", tx"s"X" = SOl , N(X") = 1 . G"
Call
the group consisting of the
g'
the subgroup of
g'
for which
9 for whi ch
X", which is of type Tm_2 ; call
X" = 1m_2 ; call
u=O; if m=2, g=g' =g". Then
product of g'
and
g'/g"; g"={e}
in the case 01; otherwise
to
G" = g/ g'; g'
0-, as the case may be; g'/g"
g"
the subgroup of
9 is the semidirect
is the semi direct product of g"
is isomorphic to
is isomorphic to
g"
and
Ga , or
Om-2. This proves
the lemma. Now we apply the lemma, and Witt's theorem, to the consideration of "spheres"; by the sphere of radius L = L(p)
defined in
Om
by the equation
p, we understand the variety F(x) = p (in the cases 01,
L2(a), S2, this is actually, in the cbssical terminology, the sphere of radius
IP).
To begin with, Hitt's theorem says that, if
vectors, other than ~'
=M~;
0, in
Lk' there isM € R'k
other than subset of
0, in
R*; put
are two
t MSM =Sand
H, as in 3.4, be
(and therefore also of every vector
O~) under
~,
L* = L ('\H; this is a Zariski-open
L. With these notations : Lemma 4.1.2 •
P€Ok
such that
for our purposes, however, we need more. Let
the orbit of the vector e
~,~'
.!i
(case 02), and
L CH, and consequently
Let
p €k
pto, and L*
(case L2(a), 01, S2), p €O; L
is the sphere of radius
(case L2(b)), p, we have
=L
K be a field containing
isomorphic to a matrix algebra
Mr(O")
k; then
OK = DK®K
is either
over a central division algebra
- 77 D"
over
K (cases 0, S, and L2 for
K:p k') or to a di rect sum
M (D")@M (D") of two such algebras (case L2 for K::>k'). In the former r r case, H is as described in 3.4; call a an isomorphism of DK onto Mr(D"); we have to show that, if matrix a(x)
since
is of rank
XE:D~
and
F(x) = P f 0, the (mr,r)-
r; this follows at once from the relation
P is invertible in
Dk , so that a(p)
consider the case L2, with
K::>k'; if a
Mr(D") @Mr(D"), the involution
x-+x
must be of rank
r. Now
is any isomorphism of DK OrID
of D, transported to the latter
algebra by means of a, must exchange the two components, since it induces the non-tri vi a 1 automorphi sm on the center choose
K@ K; therefore we can
(Y ,Z) -+ ( t Z, t Y). Put a = (a 1 ,a2 ),
a so that this involution is
where
a1 , a2 are two homomorphism of DK onto
a, a"
a2 in the obvious manner to
x of D~
sists of the elements 01(x), a2 (x) above, that
over
D"
D~ and
implies
and extend
RK = Mm(D K). Then
HK con-
such that the two (mr,r)-matrices
are both of rank
F(x)=PfO
~~r(D"),
r; and one sees, just as
x€H K•
We can now generalize \<Jitt's theorem as follows: Lemma 4.1.3. such that
(i) Let
a, a' be in
a' =Xa, tX·S·X=S. (ii) Let
l:K; then there is
G be of any type except L2(b);
also, assume, if G is of type 01 or L2(a), that m~2;
and let a, a'
~
XcR K
l:K; then there is
m~3,
or that
XE:G K such that
PfO, a' =Xa.
In the cases 01, S1, (i) is nothing else than Witt's theorem;
in the case S1, (ii) is the same as (i); in case 01, take tha t
a' = X1a, tx 1SX 1 = S; if N(X 1) = 1, ta ke
X1~RK
such
X= X1; if N(X 1) = - 1, ta-
ke X2E:R K such that X2a = a, tX2SX2 = S, N(X 2 ) = - 1 (this can be cons-
tructed by reasoni ng .iust as in the proof of Lemma 4.1. 1), and X= X, X2 • In the case S2, (ii) is the same as (i); (i) is Witt's theorem if DK
- 78 is a quaternion algebra over DK onto
M2 (K); o(a)
is a (2m,2)-matrix, of rank 2 since
which can be written as similarly write Mm(D K) onto
K; if not, there is an isomorphism
(b 1 b2 ), where
a E:H K,
K2m , and we can
b1 , b2 are in
o(a') = (b; b,p. The extension of a
of
0
to
Mm(D K) maps
M2m (K), and
ting bil inear form the assumption
GK onto the symplectic group of an alterna(Yl'Y2) on K2m x K2m; and it is easily seen that
F(a) = F(a')
(b 1 ,b 2 ) = (b; ,b 2);
is then equivalent to
our assumption is then a special case of Witt's theorem, applied to K2m
and to the subspaces of
2. The
b;, b
respectively spanned by b1 , b2 and by
proof of (i) for the case 02 is quite sim'l'.lar; in order to
deduce (ii) from this, we observe, if K, that
N(X) = 1 for all
DK
is a quaternion algebra over
XEG K (in fact, this is so for
m= 1, as one
finds by direct calculation; in the general case, it follows then immediately from the fact that
GK is generated by the "quasi-symmetries",
i.e. by the elements which leave invariant a nonisotropic hyperplane; cf. Dieudonne, Geom. des gr. class. p.24 and 41); if there is an isomorphism a
of
DK onto
M2 (K), this transforms
GK into a group of type
(01)2m' and one has merely to apply what has been said above for the type 01, observing that the exceptional cases for that type cannot occur here since
2m ~ 2, and since
p cannot be
0 if m= 1 (for a 11 types,
L:*
is empty if m= 1, p = 0). In the case L2(a), (i) is Witt's theorem
if
K~k',
and (ii) can be deduced from (i) just as in the case 01; if
K:>k', we have onto
DK=Dk®K=k'®K~K(!)K;
Kk, and
2:;' = GA/9 A; the same is true in the case L2(b) if G is replaced by the group G*= {XER*ltxSX=S} ,and g by the subgroup of that group leaving s fixed. We now consider problems (I), (II) for the groups in question. All available evidence goes to show that (II) is to be answered affirmatively (i.e., that GA/G k has finite measure) for
~
semisimple
group~
and that the answer to (I) is given by "Godement's conjecture" : if G is semisimple,
GA/G k is compact if and only if Gk contains no unipo-
tent element. (Added in December 1960 : these statements have now been proved by Borel and Harishchandra for all semisimple groups over number-fields). In the direction of Godement's conjecture, we prove: Lemma 4.1.4. Let r
be any locally compact group; let y be a
discrete subgroup of r, such that r/y is compact. Then the orbit of
!!!l s €. y under the group of inner automorph isms of r is closed. In fact, this orbit is the union of the compact sets {k-'s'klke.:K}, where
K is a compact set such that r=yK, and where
one takes for s' all the distinct transforms of s under inner automorphisms of y; and the family consisting of these compact sets is lo-
cally finite (i.e., only finitely many of them can meet a given compact
- 80 subset of r). (This lemma is, in substance, due to Selberg; cf. Bombay Colloquium on Function-Theory, 1960, p. 148-149). In view of this, the necessity of the condition in Godement's criterion will be proved if one shows the following: if G is a semisimple algebraic group, and sure of the orbit of
~
~
is a unipotent element of Gk, the clo-
under the inner automorphisms of GA contains
the neutral element of G. We do not discuss this in general. For the classical groups, it can be verified easily: Type Ll : Consider the group R(1), R=Mm(D), m~2; as M2(D)(1) is a subgroup of Mm(D)(1)
for m>2, it is enough to discuss the case
m= 2. Consider th~ ~~bit of
(6 1)
ced by elements
(~ ~).
X tS
under the inner automorphisms indu-
I k' for a sequence of va 1ues of x tend i rg
to 0 in Ak. Type (01)m' m~ 3, F not of index 0 : Take coordinates so that S is as in Lemma 4.1.1; consider the elements x-lAX, with
1- 0 b = -2 1t aSOl a, c = - t aS ", x E: I k; an d , as a bove, t ah were a e: km-2 , aT'
ke a sequence of values of x tending to 0 in Ak. Other types,
F not of index 0: As the matrix S must be
invertible, the latter condition implies
m~2;
take coordinates so that
S is as in Lemma 4.1.1; then, as for the type Ll, it is enough to discuss the case m=2. Consider the unipotent element where a!a=O, a€D k, afO
(6
~) in Gk,
(notations of Lemma 4.1.1; take aE:k',
a + a = 0, a f 0, in the cases L2, S2, and a €ok, a f 0, in the cases 02, Sl); take its transforms under the inner automorphisms induced by (fl with x E:I k, x tending to 0 in Ak.
~),
- 81 -
We now seek to prove that GA/G k is compact, for the types other than L1, whenever
F is of index
O. This will be done for all
types except L2(b); for the type L2(b), we only do it for the group
G*
defined in Theorem 4.1.1. ~ie
index
begin by considerations which are valid, whether
F is of L2(~,
0 or not. For the time being, however, we exclude the type
and also, for the types 01, L2(a), the cases and m= 1. Let
0,
O~, of measure> 1; as in the
C be a compact subset of
proof of Lemma 3.1.1, put
m=2, F not of index
C' =C+(-C). For XC:G A, the automorphism
x+X -1 x of 0Am has the module 1; therefore it maps
C onto a set
X-'C of measure> 1, which cannot be mapped in a one-to-one manner onto
O~/O~; this means that X-1C'()O~ must contain an element
its image in
~fO, so that ~=X-1c with CE.C'. Then F(c)=F(~), and, if we put P = F(~),
P is in F(C')(l Ok' which is a finite set since
compact and For each
Ok
discrete in
i, let
~i
be the sphere of radius ~i
F(x) =Pi); choose a vector torfO
in
O~, such that
if and only if leaving
~i
0A; write that set as
in
fixed, and call
4>i
by Theorem 4.1.1, we can identify
= 0,
is
P1""'P~.
(the variety
(q")k' if there is one (i.e. a vec-
F(~i) =Pi; for
F is not of index
Pi
{p o
F(C')
i =0, there is such a vector
0); let
gi
X+X~i
the mapping ~i
with
be the subgroup of G ~r;
of G into
G/g i , and then
4>i
becomes
the canoni ca 1 mapping of G onto G/9 i • Now put Ei = (~r) A() C', and 81 = 4>i 1(E i )· Our proof shows that, if Xis any element of GA, there is an 1,e"
i, and an element
c of
by Lemma 4.1.3, of the form
15 in Ei' so that XM
-1
C', such that ,
M- t,;i
~ = X- 1c is in O:Pk'
!l:!.!
Pi = F(~i)
XM
-1
~i
€. Bi' XE: BiG k• We formulate this as a lemma:
Lemma 4.1.5. There are finitely many factors
and a compact subset C'
X~=
with ME:G k ; then ~i fO
in
O~,
of O~, with the following properties: (a)
are distinct elements of
Ok; (b)
~
~i
be the sphere
- 82 of radius ping
Pi' gi
X+ XS i
the subgroup of
of
G into
l:i; put
G leaving
si
~i
fixed,
the map-
-1
Ei = (l:i) A() C', Bi = ~i (E i ); then
GA= UiBiG k• As indicated above, we choose notations so that if 0; in particular, if this notation, l:i=l:i; as
Ki
of
if 0, for all
(l:i)A
subset of
F is of index
~i(Ki):::>Ei; then
Now assume that
Ki
of
BiCKi(gi)A'
0, and that
i, so that there is, for every (gi)A = Ki(gi\; then
(gi)A/(gi)k
i, by Lemma 4.1.1,
gi
is
i, a compact subset BiCKiKi(gi)k' and, in
GA = (UiKiKi )G k , so that
view of Lemma 4.1.5, we have pact. But for each
i f 0, there is a compact subset
F is of index
(gi)A such that
if 0, we have
D~, (Ei)/'\C', is a compact
is a closed subset of
GA such that
Pi f 0, i.e., in
i. By Lemma 4.1.2, for
(l:i) A' Therefore, for every
compact for every
0, we have
Pi f 0 for
GA/G k is com-
is the group, of the same
type as G but with m-1 substituted for m, acting on the space tsiSx = 0 and leaving invariant the form induced on that space by F; obviously, the latter form is of index by induction on
0 if F is of index O. Now,
m, we can prove
Theorem 4.1.2. For all types other than L2(b), the group determi ned by (1) is such that xrJ. 0
.
~
GA/G k is compact whenever
G
t xSx f 0 for
Dm
k'
The theorem is trivially true for m= 1 in the cases L2(a), 01 (for then m= 0
G is reduced to the neutral element), and vacuously true for
in the cases 02, S2; the induction proof is val id for
L2(a), 01, and for m=2
m> 2 for
m> 1 for 02, S2 (one could also deduce the cases
of L2(a), 01, and
m= 1 of 02, S2, directly from Lemma 3.1.1).
In the case L2(b), we consider, instead of
G, the group
G*
defined in Theorem 4.1.1. One proves then, exactly in the same manner, that
Gl/G~
is compact if
F is of index
O. We observe that
G*
is
- 83 -
i sogenous to GxU, where U is, as before, the commuta t i ve subgroup of Rk, /k (Gm) determined by
zz; 1.
We sha 11 not proceed further wi th
the investigation of the type L2(b), which, in all respects. is the most difficult of all. Now we apply Lemma 4.1.5 to proving that, for all types except possibly L2(b) and 02,
GA/G k is of finite measure. Apply Lemma 2.4.1
to GA, (gi)A' (gi)k' and to the characteristic function on
(Zi)A; this shows that the image of Bi
measure if and only if
Ei
of Ei
in GA/(gi)k is of finite
(gi)A/(gi)k and Ei
are so (of course we are
using invariant measures on GA, (gi)A and seen that
fi(w)
(Zi)A). For i
of
0, we have
is compact, hence of finite measure. Proceedinq by induc-
tion on m, and using Lemma 4.1.1, we may assume that of finite measure; so the image of Bi
(gi)A/(gi)k is
in GA/(gi)k is of finite mea-
sure; as the obvious mapping from GA/(gi)k onto GA/G k is locally a measure-preserving isomorphism, this
impl~s
that the image of Bi' which
is also the imaqe of BiGk' in G'A/Gk' is of finite measure. In view of Lemma 4.1.5, the induction part of our proof will be complete if we show that Eo and
(go)A/(go)k are of finite measure. The latter fact, in
view of Lemma 4.1.1, is also a consequence of the induction assumption. Thus it only remains to show that Eo; ZAnc' the invariant measure on of radius
ZA) when C'
is of finite measure (for
is compact and Z is the sphere
0; this will be done in 4.2 for the types L2(a) (m>3),
01(m> 5), S2(m> 2); the case S1 has been treated in 3.5. The case L2(a), m.l, is trivial, and the case L2(a), m;2, has been treated in 3.7; the
cases 01, m; 3 and 4, have been treated in 3.7; the case S2, m; 1, is included in Theorem 3.3.1; therefore this will prove: Theorem 4.1.3. If G is defined by (1), GA/G k is of finite
measure for the types L2(a). 01, Sl, S2, except only for the case 01,
m• 2,
F not of ; ndex
o.
- 84 -
The same would be proved for the type 02 if we could show, also in that case, that Eo is of finite measure. As to the type L2(b), our method could be applied to the group G*, and would show that GA/Gk is of finite measure, again under the assumption that Eo
is so. These ca-
ses will not be considered any further. 4.2. End of proof of Theorem 4.1.3 (types 01, L2(a), S2). In the remainder of this chapter, we shall consider only the cases 01 (quadratic case), L2(a) (hermitian case) and S2 (quaternionic case); we put 0 = [Ok: kJ; in the quadratic case, the hermitian case, Dk = k'
Dk = k and 0 = 1; in
and 0 = 2; int the quaternionic case,
Ok is
a field of quaternions with the center k, and 0 = 4. In all cases,
O~
is a vector-space of dimension om over k, F(x) = t xSx is a k-valued quadratic form in that space,
Om is an affine space of dimension om
in the sense of algebraic geometry, and the sphere of radius
p is the
hypersurface defined by F(x) = p. In this section, we assume that F is not of index 0, and E will denote the sphere of radius 0, i.e. the hypersurface F(x) = 0 in Om; as before, we put E* = E e=(1, 0, .... 0)
n H,
where His the orbit of the vector
in Om under the group R*=Mm(O)*.
Lemma 4.2.1. For the variety E*, (1) is a set of convergence factors, provided om>4. This is done by computing the number of points of E* modulo p for almost all
p (the formulas for this are well known) and applying
Theorem 2.2.5. We exclude all
p for which S is not in
~1m(Oo)'
all
-p
p which divide 2N(S), and all
p which are ramified in
k'
(resp. in
Ok) in the hermitian (resp. quaternionic) case. Then, in the quadratic case, the number of solutions, other than 0, of F(x) = 0 in the field Fq wi th q = N(p) elements is qm-1 - 1 if m" 1 mod. 2, and (q m' -E:)(q m' - 1 +E:) with m' =m/2 and e:=:!;1 if m"O mod. 2(e:=+1
- 85 -
or
-1
(-1) m'
according as
det
(S)
is or is not a square in Fq)'
In view of Theorem 2.2.5, this proves the lemma in that case. In the hermitian case, consider first the case when p does not split in i.e. when it can be extended in only one way to
k', so that
kp
k', is a
quadratic extension of kp' and o'ip is a quadratic extension of -p Fq = 0-p Ip; then F{x) determines a quadratic form in (o' Ip)m conside-p red as a vector-space of dimension 2m over Fq , so that the number of solutions, other than 0, of F{x) = 0 modulo p is qiven by (qm_E){qm-l+E)
with a suitable £=:1. If p "splits" in
if it can be extended to two distinct valuations
k', i.e.
p', p" of k', then,
reasoning as in the latter part of the proof of Lemma 4.1.3, we see that the number of points of E* modulo p is the number of pairs of vectors 2p x, y in F~, other than 0, satisfying a relation tyS'x = 0, where S' is an invertible matrix in Mm{Fq); this is equal to
{qm - 1)(qm-l - 1).
The conclusion is the same as before. In the quaternionic case, reason1ng as in the first part of the proof of Lemma 4.1.3, we see that the number of points of rank
E~
is the number of {2m,2)-matrices (Xl x2) of -p 2 over Fq such that {x 1 ,x 2) = 0, where is a non-degenerate
alternating bilinear form on F~mxF~m; this has the value (q2m_ 1)(q2m-l_ q). The conclusion is again the same.
(£), for 0 ~ v ~ - x£v of Dm, cons i dered as an
(4) . Actually, we shall prove a stronger result (needed in the next sections) : Theorem 4.2.1. Assume that om> 4, and call
dWA the Tamagawa
measure on LA' invariant under GA. Then fL*(w)dwA4). As the same is true of TTp(1_q2-m),
where ~p
is the
this completes our proof. 4.3. The local zeta-functions for a quadratic form. Notations remain the same as above. Lemma 4.3.1. Let V be the Zariski-open set defined by F(x)
f0
in the affine space Dm of dimension
om> 4. Then
( 1-q -1 ), q = N(p), is a set of convergence factors for
V.
In fact, by the same formula which was used in the proof oj Lemma 4.2.1, the number of points of V modulo p, for ?lmost all
p,
- 89 -
is qom-1 (q-1)
is odd, and qm' - 1(q m' -E)(q-1)
if om
if om = 2m'. In
view of Theorem 2.2.5, this proves the lemma. As
V is isomorphic to the variety, in the affine space of di-
mension om + 1, with the generic point the points \~e
x €D~ such that F(x)
£
(x, 1/F(x)) , VA is the set of
I k.
now wish to calculate, for almost all
p, the following "10-
cal zeta-function" : Zp ( s) = f
(1)
V kp
IF(x) Is 1. In the v+ 1 v x ±t 0 mod. p, the measure of the subset where
x E:.( 0_p ) om,
F(x):::O mod. pV, i.e. where
IF(x)lp~q-V, is equal to
v ~ 1; and the measure of the subset where IF(x)l p =1, is
F(x)
q-omv Nv
for
$ 0 mod. p, i.e. where
q-om(qom_ 1_N1 ). This gives
Z' (s) = q-om(qom_ 1_N ) + ~ q-vs(q-omv N _ q-om(v+l)N p 1 v=1 v v+1 A trivial calculation gives the result in the lemma. As to the value of
c, we remark the following
(a) Quadratic case
(0 = 1), m even: then
residue character of t::. mod.p), where criminant of
t::. = (-1 )m/2 det (S)
(0 = 2) : then, write
F = L~xiaixi' k' = k(o)
2
a = aE.k, xi = Yi +az i , xi = Yi -azi; then, in terms of the
variables
Yi' zi' F has the coefficients
c = (am/p), i.e.
c = 1 for
m even, and
Ok
over
k, with
c = (a/p)
for
m odd.
c = 1.
ti,ui,vi,w i ,
1, i ,j, ij
i 2 = aE:k, }=bE:k, ij=-ji; if we put
F = Lixiaix V xi = ti + iU i + jV i + ijw i , then, in terms of the bles
om
ai' -aia; therefore
(c) Quaternionic case (0 = 4) : we can take a base for
is the dis-
F;
(b) Hermitian case with
c = (lI/p) (quadratic
F has the coefficients
om
varia-
ai,-aia,-aib,aiab, so that
- 91 -
4.4. The Tamagawa number (hermitian and quaternionic cases). From now on (in this section) we assume that 0 = 2 (hermitian case) or 0=4 (quaternionic case); we use "resp." to refer to these two cases (in that order). In both cases, we shall denote by the norm-mapping of 0* into Gm; its kernel is
z+v(z) =
zz
U (in the notation
of 3.7) resp. 0(1). In both cases, v maps 0A onto an open subgroup of
I k. In the hermitian case, by class-field theory, v(OA)·k* is an
open subgroup of Ik of index 2; in the quaternionic case (cf. Lemma 3.3.2), v(OA) contains all elements of finite places of k are all we define a character A= -1
>0, so that v(OA) ·k* = I k. In both cases,
of Ik by putting
>..
on the complement of that group in
is the character of Ik k'/k
Ik whose components at the in-
o~
>..
= 1 on v(OA) ·k* and
I k. In the hermitian case,
>..
order 2 belonging to the quadratic extension
in the sense of class-field theory; in the quaternionic case, A
is the trivial character of module taken in
I k. By
I I , we always denote the idele-
I k.
In the quaternionic case, we shall construct Fourier transforms ) where Xo of functions in 0Am by means of the character Xo (t-xSy,
.
1S
the character of 0A introduced in 3.1. In the hermitian case, we have DA = Ak
I,
and we do the same by means of X (txSy ), where X' i s the
character of Ak'
I
defined by Theorem 2.1.1; in both cases, we simplify
notations by writing X instead of X' Fourier transform of ~(x)
in
resp. XO. If 'I'(y)
is the
O~, defined by
(dx = Tamagawa measure in O~), and if XE:Mm(OA)
is such that t xsx = s,
then the Fourier transform of ~(Xx)
with
IS one sees by replacing x, y by
is 'I'(X 'y)
X' = S-1.t x-1. S,
Xx, X'y and observing that, for
- S2 t-XSX=S, the module of the automorphism if
x+Xx of DAm is
zC:D A, the Fourier transform of (xz)
is
1. Similarly,
Izzl- om / 2'1'(yz-1).
Our method will depend upon the construction of a zeta-function (whose residue, as usual, gives the Tamagawa number) by means of a function (x)
in D~ of which we assume that it is
"of standard type" in
a sense similar to that defined in 3.1, and also that Theorem 4.2.1 is valid both for and for its Fourier transform'!'; such functions can be obtained by the procedure described in 3.1 (following the definition of the "standard type"). We are concerned with the group
(we know that N(X) = 1 is a consequence of t-XSX = Sin the quatern i 0nic case, but not in the hermitiant case). By 3.7(b) resp. Theorem 3.3.1, we know that T(G) = 1 for om=4. From now on, we assume om>4. If V is as in Lemma 4.3.1, we denote by V; the open subset of VA given by >..(F(x)) = 1. With this notation, we introduce the function (1)
where d'x
is the Tamagawa measure on VA derived from the gauge-form
dx 1... dx om
(if the Xi
are the coordinates of x for any choice of a
basis of D~ over k) and from the convergence factors I~e
(1_q-1), q=N(pl
put, for v = 0, 1 (hermitian case) and for v = 0 (quater-
nionic case) :
- 93 -
then we have Z¢(s) = calculation for
~(Io+I1) resp. = 10 , We give now a multiplicative
10 , 11 resp. for
10 ; this is similar to the correspon-
ding calculations in Chapter III; Iv
is the product of a "finite part"
(i.e.,of an integral over a finite product
TTVES
of "local zeta-functions"
Vk ) and of a product v
For v=O,thisis given by Lemma 4.3.2; for v = 1, Ap character induced by a subgroup of Ap(t) = A(p)r
k*p considered (in the obvious manner) as I k; if p is not ramified in k I, th is is given by
if
of
k'
such valuation
A on
Itlp ={, with
"spl its" or not in p', p"
is the local
A(p) = +1
or
-1
according as
p
k I (i .e. according as there are two valuations
extending p', with
p, with k~1
k~I" k~""
kp'
or there is only one
quadratic and non-ramified over
kp)' But
then we have
s+~ log q
so that Lemma 4.3.2 gives the value of the local zeta-function also in this case. In the quaternionic case, we find (in view of the remarks following Lemma 4.3.2) that the infinite product for almost all
I
o
coincides, for
p, with that for
which shows that it converges absolutely for Re(s)
>0,
and that
- 94 Similarly, in the hermitian case, the infinite product for same (for almost all
(m even), (m odd),
I;k(s+1)L k '/k(s+m)L k '/k(m)-1 Lk'/k= I;k,/I;k
tension 11
k'
of
is the
p) as that for . I;k(s+1 )l;k(s+m)l;k(m)-1
where
10
is the L-function belonging to the quadratic ex-
k, i.e. to the character
A. The infinite product for
is the same as that for (m even), (m odd).
As
m> 2 in this case, this proves
the absolute convergence for
Re(s) > O. Furthermore, we find that
Thus, in all cases, the integral for for
Z~(s)
is absolutely convergent
Re(s) >0 , and
(2)
Now we give the additive calculation. Take any x€V;; this means that
XED~
z€D A, i.e. that
and that
F(x)
is of the form
zpz
with
pEk*,
F(xz- 1) =p. By Hasse's fundamental theorem on quadrati:
forms, the fact that the equation
F(x')=p
implies that it has a solution
in
E;;
Vk' whi ch is the set of the vectors the equivalence relation
has a solution
D~; then we have E;;
€'D~
F(E;;')/F(E;;) =~I;
such that with
x'
in
D~
F(x) = F(E;;z). On F( E;;)
+0,
cons i der
l;eD k; by Lemma 4.1.3,
- 35 -
two vectors only if S'
S'S' =
are in the same equivalence class for this if and 1;; C:D k. Let
Me::G k,
Mt;1;; with
(t;f.!)
representatives for the equivalence classes on put
Pf.! = F( sf.!); for
f.!
f v,
pi Pf.!
be a complete set of Vk under this relation; ~1;;
cannot be of the form
with
1;;EDk; therefore (by the norm theorem for cyclic extensions, applied to k'/k
in the hermitian case, and by Eichler's norm theorem in the qua-
ternionic case)
p /p cannot be of the form ZZ with z6D*A. From v f.! this, one concludes at once that, for every x<sV;, there is one and only one of
f.!
such that
F(x)
=
zpf.! Z
+
VA where this is so for a given
with
ZED A *; let r2
be the subset
f.!
f.!; this is an open subset of
VA'
and we get : Z0,
and it gives the value of the right-hand side, showing in residue~
particular that it has the this with (2), we get ,(G) 2.4.2 to
:A'
=
,(G)'I'(O)Pk for
s = 0; comparing
1. In the general case, we apply Lemma
which, by Lemma 4.1.4, we may identify with GA/g A, where
g is the subgroup of G 1eavi ng some vector So E.: k fi xed; by Lemma 4.1.1 and the induction assumption, we have ,(g) = 1. This gives:
~(w)
replacing here
by
~(wz),
and applying the same formula to
'I'(wz- 1), we get (since the automorphism w+wz of !om-1 ges dW A into Izzl2 dwA):
Z~(s)
(5)
:A'
for
z cD A, chan-
Z~(s) + z!(-s-~om) + Pk,(G)('I'~O) - till) s+~om
=
in the number-field case, and a similar formula, which we omit, in the function-field case. Since pk'l'(O), we get ,(G) i.e. when than
=
(2)
shows that the residue at s = 0 must be
1. One may observe that, when GA/G k is compact,
F is of index 0, the zeta-function has no other residue
1 s = 0, s = 20m
(and these residues gives the value of the Tamagawa
number) while otherwise it also has poles at s = -1, s = 1 - ~om; this should be compared with similar results in 3.8 for the zeta-functions of simple algebras. In the hermitian case, let again G* if we take coordinates so that
S appears as
be the group
(~ ~r)
{txsx = s};
, with
a E:. k*
- 99 -
and S'€Mm- l(k'), we see that G* is the semidirect product of G and of the group [(~ ~m-,) IZl = 1] , which is isomorphic to U. Now we have seen that r(lJ)
is isomorphic to a group G,*={tx's'X' =S'}; if
is the subgroup of G'*
G'
r(lJ)
determined by det(X') = 1, we see that
is isomorphic to a semidirect product of G'
induction assumptlon,
(~)
and
and, by the
is a set of convergence factors for G', and
'r(G') = 1. As we ilave seen in 3.7(c) that (1_t..(p)q-1) vergence factors for
U;
U, it is also such for
is a set of con-
r(lJ); and the measure of
rilJ)/r~lJ), for the Tamagawa measure derived from those convergence factors, is T' = 2Pk,/Pk' since it has been shown in 3.7(c) that this is so for
U. As we have seen that
for
V, we conclude from this, as above, that the factors
are convergence factors for
(1_q-1)
is a set of convergence factors
r = Gx D*; as they are such for
D*, thi s
shows that (1) is a set of convergence factors for G. Using these sets of convergence factors, we can again apply Lemma 2.4.2 to rA' ri lJ ), ~lJ' r k, r k(lJ) ,and get a formula similar to (4), except that Z¢(s) has now to be replaced by T'Z¢(S). The continuation of the calculation is just as before, except that the application of Theorem 3.1. l(iii) to D* troduces now the constant Pk'
in-
instead of Pk. Thus (5), or the corres-
ponding formula in the function-field case, will be valid provided we replace T'
T
'Z¢ ,T 'Z¢+'
T 'Z¢ _,
= 2Pk,/Pk' this means that (5) is valid in the hermitian case if Pk
is replaced by
~Pk. Comparing this with (2), we get T(G) = 1 as before;
and the result is the same in the function-field case.
Theorem 4.4.1. We have T(G) = 1 for the group G defined by (1)
~
ill.E!) •
4.1, in the cases
L2(a) (hermitian case) and S2
(quaternionic
- 100 Remark. The group
f,
acting on Dm by x .... Xxz. is not effec-
induces the identity on Dm if z is in Z* , where Z is the center of D, and X=z -1 ·1 m; the condition N(X) = 1 gives then z4
in the hermitian case, our methods can also be ap-
plied to the following group f'
if
= {(X,].J)c: Mm(D)* x Gmitxsx = ]1S, det(X) = ]1m/2} ;
(X,]1)C: f', X is called a similitude of multiplicator ]1; if Mk, MA
are the sets of the multiplicators belonging respectively to the element of
fk
and
fA'
it follows from Dieudonne's theorems on similitudes
that Mk = MAnk*, and that MA consits of the ]1 E;1 k such that ]1v > 0 for every real infinite place v of k for which (i) kv=8,
k~=~
for
wiv, and (ii) F and -F are note equivalent as hermitian forms over kv. Using this, it can be shown that, when we write 2Z0, and that only the one corresponding to A= 1 gives a re-
sidue for
s = 0, this residue being PkJ(x}dx. Therefore the integral
for Zs
is absolutely convergent for
Re(s} >0, and we have:
(4)
Now we can also write HSk* as the disjoint union of the sets HSp when we take for
P a complete set of representatives of k* modulo
k*()H S' i.e., under our assumptions on S, modulo
k*2; therefore, if we
put (5)
these integrals are absolutely convergent, and we have (6)
this series being also absolutely convergent for
Re(s}
>0.
Now consider Z(s,p}Zs(s,p}-1; the multiplicative calculation shows that this is the product, extended over all valuations of the factors
v of k,
- 107 (7)
with Hv
=
k~2 if ve:S, Hp = kp2Up if p tS. For
VIS
S, T(p), which is
the subset of the variety F(x) = pi determined by y of 0, covers twice the subset of A~ determined by F(X)t:pk~2; therefore the above factor up€ Up - u~, so that 2 2 U = u Uu u and H = k*2 Vk*2 u . then the second integral in (7) is p p PP P P P p' the sum of the same integrals taken under the restrictions F(x)€ k~p, has then the value 2. For pitS, take
F(X)E k~Upp, respectively; and these, for the reasons just explained, differ from the same integrals, taken over T(p) 1
the factor 2. Thus the factor (7), for
By Lemma 4.5.2, this is equal to
and T(u pp), only by
p¢S, has the value
for m even; in this case, there-
fore, the integral in (1) and the series in (2) are absolutely convergent for Re(s)
>0; and we have
For m odd, Lemma 4.5.2 shows that the factor (7), for PISS, has the value
1 whenever p contains an odd power of p; when that is not so,
this same factor has the value -m'lt -2s-2-m ' ) e(p) = ( 1+ng 1-ng 1_q-2S-m-1
wi th
n =! 1; ina" cases, therefore, we have 1 - q-m I
<e(p) 4, we have here m'
~
2. This shows that the infinite product for
is absolutely convergent together with that for
Z(s,p)
ZS(s,p); also,
if we put 11 (S)
=
TT
(l-q -m' ) , 11' (S) =
~S
TT
we see that 2-I s 1Z(s,p)ZS(s,p) -1
is > Il(S)
this implies that the series (2) for Z(s) and that 2- ISI Z(-s)ZS(s)-1
and
4. This completes the "multipl i-
cative calculation". Now we take up the additive calculation. Take any p such that T(p)
is not empty, i.e. (by Hasse's theorem) such that there is
t;;oE km for which
F(t;;o) = p. Put
f
= Gx Gm, and let
f
act on T(p)
by
((X,t),(x,y))+(Xxt,yt) ; by
~Jitt's
theorem (Lemma 4.1.3),
subgroup leaving f'
= G'
x {1},
(t;; o ,1)
where G'
f
acts transitively on T(p), and the
fixed has a cross-section; this subgroup is is the subgroup of G leaving t;;o fixed. By
Lemma 4.1.1 and the induction assumption, (1) is a set of convergence factors for
G', and
T
(G' ) = 2. Let dX, dX'
for G, G'; then dX'(dt/t)
be i nvari ant gauge-forms
is such a form for f. Clearly
y-mdx1 ... dxm is a gauge-form on T(p), invariant under f. Therefore,
- 109 -
by Theorem 2.4.3,
r
has the same set of convergence factors as T(p),
viz. (1_q-1), so that G has (1) as a set of convergence factors; also, the Tamagawa measures for r,r' ,T(p), derived from these convergence factors and the gauge-forms dX'(dt/t), dX', y-mdx , match together topologically. This gives, by Lemma 2.4.2 :
where the sum is taken over all and all
(M,T)c: rk/r k, i.e. over all
M.sGk/G k
T £ k*; but then, by Witt's theorem, the vector i; = Mi;o T runs
twice through the set of all vectors in
km such that F(i;)EPk*2; if
then we let p run through a full set of representatives of k* modulo k*2, F(i;)
i; runs twice through the set of vectors i;
f O.
in
km such that
This gives
From here on, the calculation is exactly the same as in 4.4, beginning with formula (4) of that section. The conclusions are the same; in particular, the residue of Z(s)
at s = 0 turns out to be PkT(G)/2; com-
paring this with (8), we get: Theorem 4.5.1. The Tamagawa number of the orthogonal group in m~
3 variables is 2. Remark. For indefinite quadratic forms, Siegel has defined ZetT
functions for individual classes of such forms. This can perhaps be explained by the fact that, for indefinite forms, classes and "spinorgenera" are the same (Eichler-Kneser). If G is the orthogonal group,
- 110 and ~ is the corresponding spin-group, the spinor-norm, for an element of GK(K=any field) is an element of K*/K*2 which gives the obstruction against lifting that element from GK to defines a homomorphism of GA into therefore, to every character of
~K.
In particular, this
Ik/I~k*, with the value 1 on Gk;
Ik/I~k*, i.e. to every character of
Ik belonging to a quadratic extension of k, one can assign a character of order 2 of GA, with the value
1 on Gk; it is not unlikely that,
by introducing such characters into our zeta-functions, one might get Siegel's zeta-functions for indefinite forms.
- 111 -
THE CASE OF THE GROUP G:z by M. Demazure
The method used in the case of orthogonal groups can also give the Tamagawa number of the groups of type G2, which turns out to be
1
as expected. We first recall some results on Cayley Algebras, after JACOBSON, Composition Algebras and Their Automorphisms, Rend. Palermo, 1958. For the time being,
k is any field of characteristic not 2.
A Cayley algebra over k is a vector space 8 over k, together with a k-l inear map
~k
of dimension
denoted
~k x ~k -+~k
(x,y) -+ x· y and a non-degenerate quadrati c form ca 11 ed the norm N :~k -+ k subject to the following axioms: (i) there exists in
~k
a unit-element, i.e.
1E:~k
with
x·1=1·x=x; (ii) for any x, y E~k' N(x·y) = N(x)N(y). One can easily show that the form N is uniquely determined by the structure of (non-associative) algebra of Let
~
~k.
be the algebra-variety defined by
~k.
We denote by
~o
the orthogonal space, for N, of the one-dimensional line k·1. For x~~o'
N(x)1 = -x·x. An automorphism of
~
is a linear mapping g:
such that g(x·y) = g(x)·g(y). Then g(1) = 1 and
g(~o) =~o.
If
~-+~
XE~o'
then N(g(x)) = N(x). Moreover, one proves (Jacobson, Theorem 2) that g
- 112 is a rotation, i.e. of determinant 1. Hence the group G of all automorphisms of
~
SO{~o,N).
is imbedded in
It is a semi-simple algebraic
group defined over k which becomes isomorphic over
k to the group
G2 of the Cartan-Killing classification. A Witt-type theorem is true for
Let
(~k,N)
and
(~k,N')
G (Jacobson ~ 3) :
be two Cayley algebras with equivalent norms
(in the sense of quadratic forms). Let B (resp. B') subalgebra of
~k
(resp.
~k).
be a non-isotropt
Let there be given an algebra-isomorphism
f : B+ B'. Then f can be extended to an i somorphi sm of Corollary . .!i x,
ye:~k'
~k
onto
~k.
x, YfO, N{x) =N{y), then there exists
g EG K mapping x on y. Proof: 1) If N{x) = N{y)
f 0, then K{x) and K{y) are two
isomorphic quadratic fields. 2) If
N{x) = N{y) = 0, then x and y can be imbedded
in two quaternion algebras isomorphic under a map carrying x into y. \~e
finally have the two following results:
(i) Let
a E. ~ o , .a· a = b·1
f O. Let K= k(a). Then the orthogona 1
L of K is a 3-dimensional vector-space over K. The subgroup of G leaving K point-wise fixed is isomorphic to the unimodular unitary group of L as a vector space over K relative to the form (x ,y)
+
b-1 a (ax ,y). (Jacobson, Theorem 3.). (ii) Let B be a quaternion subalgebra of
~.
The subgroup of
G leaving B point-wise fixed is isomorphic to the multiplicative group of elements of norm
in B.
From now on, is a number field. (In the case of a function field of characteristic not 2, everything is valid, provided that we prove that Gk is Zariski-dense in G). A careful analysis of 4.5 shows that what was proved there amounts to the following:
- 113 Let rlo be a finite-dimensional vector-space (of
dimension~
5)
defined over k, with a quadratic form N. Let G be an algebraic group of rotations of N defined over k. Suppose that G verifies a Witttype theorem (i.e. if x, YErl o ' N(x)
=
N(y), there exists gEG car-
rying x into y; if x and yare rational over k, then g may be taken rational over k). For aE'rl o ' let G(a)
be the isotropy group of
a in G. Assume the two following properties (i) There exists a finite number T such that for a€rl k, the Tamagawa number of G(a)
non-isotro~c
is finite and equal to T, indepen-
dently of a. (ii) For any isotropic aErl k, the Tamagawa number of G(a)
is
finite. Then the Tamagawa number of G is finite and equal to T. In our case,
G satisfies a Witt-type theorem. We have only to
verify properties (i) and (ii). (i) By property (i) reca 11 ed above, for non- i sotropi caE rlk ' G(a)
is a unimodular unitary group and T(G(a» = 1. (ii) Let now a be isotropic. The subgroup of G leaving the
line
k·a
fixed has at most rank two. It contains a multiplicative fac-
tor not contained in G(a). Hence G(a) has at most rank one. On the other hand, let B be a quaternion algebra containing a. Then the subgroup of G leaving B pointwise fixed is semi-simple of rank one and contained in G(a); its Tamagawa number is one (by property (ii) recal-
led above). By the general properties of algebraic groups, it must be normal in G(a). The quotient being of rank zero is unipotent, hence has Tamagawa number one. On the other hand, the quotient being unipotent,
the fibration admits local cross-sections (Rosenlicht). By Theorem 2.4.4, the Tamagawa number of G(a) is finite and equal to 1. This proves: Theorem. The Tamagawa number of a group of type G2 is
1.
- 114 -
APfBIDIX 2.
A SHORT SURVEY OF SUBSEQUENT RESEARCH ON TAMAGAWA NUMBERS by Takashi Dno
The following is a short survey of works on Tamagawa numbers which have appeared since the original publication of Adeles and Algebraic groups, The Institute for Advanced Study, Princeton, N.J., 1961 (Notes by M; Demazure and T. Dno). In the sequel, we shall quote these notes as
[AAGJ. In the bi bl i ography at the end of thi s survey we have
also included some earlier work, chiefly by Siegel and Weil.
§ 1. Definition of T(G) for unimodular groups ([AAG, Ch.I, Ch.II, Ch.III, 3.6J) Let
k be an algebraic number field,
algebraic group defined over
G be a connected linear
k and w be a left invariant differen-
ti a1 form of hi ghest degree on G defi ned over k. The group Gis cal· led unimodular if the form
w
is also right invariant.
I~e
have the fol-
lowing chains of containment : unimodular
~ . t en t unlpo
I
Ga
"';eductive t orus /'" "".. 1e semlslmp
I
Gm
I
simply connected
where Ga , Gm mean the additive group and the multiplicative group,
- 115 A
respectively. We now define the Tamagawa number T(G). Let G be the group of rational characters of G, group of
G of 1
G= Hom(G,G m),
Gk
and
be the sub-
characters defined over k. Let GA be the adele group
and let GA= {XEGA,
1~(x)IA=
1 for all
•
~EGk}.
1
Then GA/G A is iso-
morphi c to the vector group fR r , r = rank (\. As a measure on GA/Gl we take the usual measure of ~r which we denote by d(GA/G1). Since Gk is discrete in GA, we define dG k to be the canonical discrete measure. Thanks to the fundamental result of Borel and Harlsh-Chandra
[1J
(see
also Borel
[1]) the space Gl/Gk has a finite measure. We can then de-
fine T(G)
as the measure one has to give to the space
in order
that
where
dG A is some canonical measure on GA to be determined. Now, ta-
ke a finite Galois extension
K/k
so that
G= GK.
Then
G becomes a
l-free Gal (K/k)-module and we denote by XG the character of the integral representation of Gal(K/k). The Artin L-function L(s, XG)
has a
pole of order r at s = 1. We put dG A= p- 1 1 I-dim G/2 TTw TT L (1 X)w G uk v p ' G p' v/oo p A
where
PG = lim (s-1)rL(s, XG)' /:'k = the discriminant of k. (As for the 5.... 1
convergence of dG A, see the beginning of Ono
(71).
It can be shown
that dG A is instrinsic, i.e. it is independent of the choice of K/k and w. Thus, the definition of the Tamagawa number
is settled. Note the properties
- 116 for any finite extension
K/k. The definition of the number T(G)
is
chosen so that T(Ga ) =T(G m) = 1. The latter equal ity is equivalent to the classical formula for the residue at s = 1 of sk(s). §2. The weil conjecture. The statement
(W)
T(G)
=
1 when G is simply connected,
is known as Weil 's conjecture. One cannot find [AAGJ; however, it is stated in Weil
(W)
explicitly in
[4J. The conjectClre
(W)
has been
settled for almost all simply connected groups. Among absolutely almost simple groups,
(W)
is not yet settled for groups of type 304' 604'
E6 , E7 and ES. On the other hand, groups (l~eil
[9]
and Mars
(W)
is settled for all classical
[2]), for the spl it groups (Langlands
and even for the quasi-split groups (Lai
[fJ,
[2J).
(W)
[lJ)
was also
settled for groups of type G2 by Oemazure ([AAG, Appendix]), for groups of type
F4, and some groups of type
1E6 (Mars
[11). We shall
later discuss the situation for classical groups and quasi-split groups in a little more detail. There are many survey papers refering to T(G) and/or
(W)
Mackey
[1J,
\~eil
[3J
(Borel
[2], Cassels
[2], Mars
[3], Ono
[1], Iyanaga
[1], Kneser
[1],
[7]). Near the bottom of p.311 of
one finds the following passage in allusion to
(W)
after a
description of Siegel's theorem on quadratic forms: ............. "Est-il possible d 'en donner un enonce general, qui permette d'un seul coup d'obtenir tous les resultats de cette nature, de meme que la decouverte du theoreme des residus a permis de calculer par une methode uniforme tant d'integrales et de series qu'on ne traitait auparavant que par des procedes disparates ?" As is well known, it was Tamagawa's discovery that T(SO(f)) =2 (Siegel's theorem on the quadratic form f) and that T(spin(f)) the late fifties, which stimulated the work discussed in
[AAG].
=
1 in
- 117 53. Tamagawa number of tori and relative Tamagawa numbers. In and T(G')
[AAG, Ch.III, 3.6J, Weil employed a trick to compare T(G) when G,G'
are isogenous. A typical case is;
G= the pro-
jective group of a division algebra of dimension n2 over its center, G' = the special linear group of the division algebra. One first proves that T(G) = n. Then, the trick impl ies that T(G') = 1, a solution of (W)
for G'. One can describe the trick in a more general setting as
follows ,,; let G be semisimple and let ring group of Gover k. Then ker the isogeny
TI
(G' ,rr)
be the universal cove-
(the fundamental group of G) of
is endowed with a structure of a Gal (k/k)-module, which
TI
can be imbedded in a torus T = (R K/ k Gm)r
for some finite extension
K/k. One has a commutative diagram with exact row and column ; 1 ~
I~
->- G' ->- G* ->- T' ->- 1
~i .1. where G* = (G'
x
T)/Ker 'IT with the diagonal imbedding of Ker'IT . Using
G* as a dummy, one reduces the computation of the ratio T(G)/T(G')
to
the i sogeny of tori ; T->- T'. As for a torus T, the Tamagawa number can
be written in terms of the Galois cohomology (Ono
[3J);
T(T) = # Pic T # W(T) where, for any algebraic group A, Pic A= Picard group and
W(A) =
Ker(H1(k,A)->-lTH1(kv,AP. Note that Pic T=H 1(k,T) for a torus T. Finalv ly, we arrive at a formula which expresses the ratio T(G)/T(G') in terms of the Gal (k/k)-module F = Ker 'IT (Ono
[4],
[7])
- 118 T(G)/T(G') =
/\
where F = Hom(F, Gm). Therefore, if T(G)
(W)*
=
# #
o
"-
H (k, F)
W (F)
(W)
holds for G', we have
o
A
#H (k, F)
#
W (F)
a conjectural formula for any connected semisimple group G.
g 4. Tamagawa number of classical groups ([AAG Ch.III, IVl) The key idea of the use of the Poisson summation formula for the calculation of the Tamagawa number of classical groups goes back to the paper Weil
[2J
where the formula is appl ied to determine the volu-
me of SL(n,IR)/SL(n,1'). When Siegel
[7J
improved an inequal ity of
Hlawka on SL(n,R), he obtained without Poisson summation the value ~(2)~(3)
\~eil
... ~(n)
[2J
for that volume, when measured in the nice fashion.
gives a simpl ification of Siegel
of the book Weil
[7] from the point of view
[1J.
In 1964-65, two important papers appeared in the Acta ca
(Weil
[8],
Mathemat~
[9]). Pushing his way in the direction of Poisson sum-
mation, Weil obtained there a formula in the framework of adeles and algebraic groups which is a wide generalization of Siegel's work on indefinite quadratic forms (Siegel
[8J,
[9J). As an application of this
Siegel-Weil formula the Tamagawa number of all classical groups except the groups of type Later Mars
[2J
L2(b) ([AAG, p.76]) were determined systematically.
took care of this last case and
(W)
was settled for
all classical groups. §5. Tamagawa number of quasi-split groups. Let us begin with the case of the Cheval ley group. Let G be the Chevalley group, i.e. the identity component of the group of
automo~
- 119 -
phisms of a complex semisimple Lie algebra g (Cheval ley
[lJ). With
respect to a Cheval ley basis of g, G becomes an algebraic matrix group defined over Q. Let w be a left invariant form on G of the highest degree defined over Q which is obtained by taking the wedge product of l-forms dual to vectors in the Cheval ley basis. Let groups of G whose coordinates are in
qR
be the Haar measure on
~,l,
be the sub-
~,G~
respectively, and let
~
derived from w. As an application of his
theory of Eisenstein series (Langlands
[2J), Langlands
proved the
[11
remarkable equality: £,
f
~/Gl
n~ i=l
w/R = # F
(a i )
where F = the fundamenta 1 group of G= Ker 1T (i n § 3) and the integers a i £, 2a.-l appear in the Poincare polynomial IT (t 1 + 1) of the maximal comi=l pact subgroup of G. Obviously, this is a generalization of the Siegel's result for the volume of
SL(n,~)/SL(n,~)
mentioned in §4. Combining
Langlands' result with a computation of the volume of GZ
for any pri-
p
me p (Ono
r5J)
and the formula of the relative Tamagawa number in
one settles
(W)
for the universal covering group of G. Later in 1974,
Lai
[1]
§ 3,
verified that the method of Langlands works for any quasi-
split group over k, as well. Namely, let G be a connected semisimple quasi-split group over k, let B be a Borel subgroup of G defined over k, and T be a maximal torus in B defined over k. The basic observation of Langlands is T(G) = (1,1) = (f,l)(l,g)/(Pf,Pg) f,g€. L2 (GA/G k) where
for
P means the orthogonal projection onto the space
of constant functions in
L2(G A/G k). Using the theory of Eisenstein se-
ries, Lai computes the terms on the right hand side with suitable f,g and obtains
T(G) = CT(T)
where c is the index of the lattice of k-
rational weights of G in the corresponding lattice for the universal
- 120 -
covering group of G. Combining this with the formula of T(T) one settles
(W)
in 93,
for all quasi-split groups. As far as we know, this
is the most general result about T(G)
obtained without appealing to
the classification theory. ~
6. Remarks (1)
Unlike
[AAG] , we have totally ignored the function field
case in this survey, because most of subsequent papers have treated the number field case only, and also because the function field case has not matured to the level of the number field case (See, however, Harder [2]). (2)
The use of the Gauss-Bonnet theorem for the computation of
the volumes of the various fundamental domains goes back to Siegel (e.g. Siegel
[5J). There are interesting relations among Tamagawa numbers,
Bernoulli numbers, Euler numbers and special values of L-functions. (See, Borel
[4], Harder
Satake
[1], Serre (3)
[1], Harder and Narasimhan
[1}, Ono
[5J,
[6],
[2]).
It is desirable to extend the notion of the Tamagawa num-
ber to a more general category of algebraic varieties defined over k. Birch and Swinnerton-Dyer considered the case of elliptic curves and produced a very plausible conjecture (See, Cassels Dyer
[1), Tate
[1]). Recently, Bloch
[1J
[1], Swinnerton-
obtained a purely volume-
theoretic interpretation of the Birch and Swinnerton-Dyer conjecture. It is interesting to note that the generalized Tamagawa number still has the form 'r(X) =
# Pic(X)tors #
W (X)
for some commutative group variety. (See also Sansuc
[1]).
- 121 BIBLIOGRAPHY
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[1]
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~]
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