Adeles and Algebraic Groups (Progress in Mathematics)

•.. dx an , where <pa(x a ) and

are eve-

- 18 -

rywhere defined rational functions, hence polynomials. We recall the defirntion of reduction of an ideal modulo p. Let

-;0

be an absolute 1y pri me idea 1 ink [V}, V= (V 1' ... , Vn)' Let us taP~(V)

ke a finite set of generators

where all the coefficients of the

of

jJ.

For a finite place p

P 's are integral, the reduction ~

p~(p) of P~ modulo p is defined and we define j9(p) as the ideal genera ted by the P~ (p)

in

rators, then for almost all as above

(;Pp)

(Qp/p) [VJ. I f we ta ke another set of genep, the reductionJO(P)

will be the same

is "almost intrinsic"). A well known theorem of

E. Noether (Math. Annalen 1923) asserts :

Theorem 2.2.2. For almost all p,Jb(p)

is absolutely prime.

Let now S be a finite set of primes such that for all

p not

in S:(1) all thef

~p) (BeA) are absolutely prime

(II) all the polynomials D(a,S) p(a,S) F(a~S) ~ ~-1 A A- 1 v 'v ' V , J ' a' a ''j'a,'j'a have integral coefficients. Then for all riety V(p)

p not in S we shall define a non-singular va-

defined over the field Qp/p = Fq (finite field with q = Np

elements) called the reduction of V modulo p. Let V (p) be the variety defined by 1o(p)e. F [XcJ. Let r(p) ( ) a q ( Sa a be the birational correspondence between Va p and Vs p) defined

bYJ5~~~S}' By the hypothesis (II), the polynomials D,P,F can be reduced modulo p, and the equations (1), (2), (3) remain true. Hence we can glue together the varieties va(p) find an abstract variety V(p) Moreover the

~

a

by the correspondences r~~) , and

defined over Fq .

are p-adic units and the n first coordina-

-(p) tes are uniformizing parameters on V a

which is therefore non-singu-

lar. 2.2.3. We want now to interpret the rational points of

V(p)

- 19 -

over Fq . We have already defined [V; f ~ VaJl Q = Ua f a (V aI Q ); we dep p (though it is only "almost intrinsic"), and we innote it here by V Qp troduce an equivalence relation on Vo • Let ]1 be a fixed integer> o. -p We say that a == a (p]1) a ,a E. Voi_f one can find an a tEA -p such that a = fa (a a ),a" =fa (a"a ) ,a a ,a"a E. Va10_p ' aa == a"(p]1). a To justify this definition we have to prove N

I

,

I

Lemma (a )=fS(a S)' aae:va/Q ' aSEvS/Qp' and - 2.2.1. -If a=faa p if a = f (a ") with a == a (p]1) -=t:..:.:he=-:n~o~ne=---=c:..::a::..:n_f~i~n::::.d aScv S/Q a a a a p such that a" = fS(a S) and a s ==a S(p]1)· I

I

As

(aa,a S) is in the graph of fS,a' we have

In view of (II), at least one of the D(a,S)(a) must be a p-adic unit; v a then, by (2), we have, for this value of v:

But, if we replace aN by ~

<~

F(a~S) (a ") and define a" . = V,J a , then S,J D(a,S) (a ") V

o(a,s) (a ") \I a

is still a p-adic unit,

a

and as is in

We are now ready to prove: Theorem 2.2.3. The reduction modulo p gives a one-to-one-correspondence between the equivalence classes modulo p of Vo rational points of ij(p)

over F.

--

and the

-p

q

Two equivalent points lie on the same affine Va by

definitio~

hence it is sufficient to prove the theorem for an affine V. In this case, V is given by an absolutely prime ideal fu . Obviously a zero of

~ in (Qp)N reduces modulo p to a zero of

fo (p).

The equi va 1ent

- 20 points modulo p reduce to the same zero. It remains only to prove the following generalization of Hensel's lemma Lemma 2.2.4. Let

kp be a p-adic field and V an affine va-

riety of dimension n defined over kp by an ideal pck p [X 1, ... XN]. Let

a

be a zero in FqN(q = N(p))

there exists

of the reduction

F1, ... FN_n ..!!!.j<>nQpIXI

Suppose that

such that the matrix

is of rank N - n. Then there exists a zero a of cing modulo p to

'fo (p).

ju

II
..!!!. (Qp)N redu-

a.

(I tis not necessary, for the 1emma to be val i d, that

f( p)

For a proof, see P. Samuel, Int.

should be an absolutely prime ideal). Congress Amsterdam 1954, vol. 2, p. 63. Let us now compute

We decompose Vo -p

into equivalence classes

w = r. f w Vo p _ Jp) x:: a(p) p

f

aE V

-p

But {xlx::a(p)}

is by definition in the image of one affine Va; on

Va' w can be written as ¢a(x)dx a1 ..• dx an . But ¢a is a p-adic unit; therefore, for a in Va/Q p f

x::a(p)

wp

=

f

x::a(p)

The projection of Va on the space

dx 1'" dx n a a

(x a1 ' ..• xan ) is an isomorphism on

the subset x:: a (p); therefore f

x::a(p)

-n dx 1" .dx n = q a a

- 21 This completes the proof of Theorem 2.2.5. If V is non-singular and w is a gauge-form on V, then, for almost all p,

V reduces modulo p to a non-singular

variety which has qn f wp rational points over Fq , where q=N(p). Vo -p 2.3. The global measure and the convergence factors A set (AV) of strictly positive real numbers indexed by the valuations v of k is called a set of factors for k. Let V be a non-singular variety, defined over k, and w a gauge-form on V, rational over k. For each place v of k, we have defined the positive measure Wv on Vk • For every p, we put 11 p(V) = f

wp.

~

v

Definition. A set of factors

(AV)

-p is called a set of conver-

gence factors for V if the product lTp(A~111p)

;s absolutely conver-

gent. This definition is valid because Vo is almost intrinsic -p If Wi is another form, f wp = f wp for almost all p. Vo Vo -p -p (This follows from Theorem 2.2.5. Alternative proof: Wi = fw

(1) (2)

where f

is a morphism of V into Gm• For almost all

f(V o )C(Gm)o =U p and -p

-p

Definition. If (AV)

Iflp=n.

is a set of convergence factors for V,

the Tamagawa measure n = (w, (AV)) on VA

it

p,

derived from w by means k

(AV)

is defined as the measure on VA

inducing in each product k

the product measure 11kdim V lTv(A~1wv) (11k is v r'_p the constant introduced in 2.1.3).

TI\IIIS Vk xTIp'!'<: Vo

Theorem 2.3.1. If ce.k*,

(W,(A V)) = (CW,(A V)).

- 22 In fact,

(cw)v = Iclvwv and TIv1c1v = 1.

Let now K/k be a finite and separable algebraic extension. Let V be a variety defined over K and

(W,p) = RK/k(V) (1.3). If w

is a differential form on V defined over K, we define the differential form p*(w) (oi)

on W as follows. Let e be such that

K= k(e). Let

be the isomorphisms of K into k and let !.ii:=TTi<j(e

0· 1 -

0·

e J).

Then we define p*(w) = (!Zi)dimv"o(po)*wo. We see easily that p*(w) is defined over k. Then one has Theorem 2.3.2. Let (W,p) = RK/k(V). Let

(A.w)

K/k

be finite and separable and

be a set of factors for

K and Ilv =TIw/v)V

Then (1)

(2)

(llv)

is a set of convergence factors for W if and only if

(Aw)

is a set of convergence factors for V.

If (1) is satisfied,

(W,(A W))

corresponds to

(p*(w),

(llv»

in the canon i ca 1 isomorph i sm VA r:::t WA (Theorem 1. 3.3) . K k This follows at once from Theorems 1.3.2 and 1.3.3 and from the definition of W. This theorem gives the reason for the introduction of the factor Ilk

in the definition of the Tamagawa measures. 2.4. Algebraic groups and Tamagawa numbers By an algebraic group, we shall always mean in this course an

algebraic group isomorphic to a subgroup of a linear group. If G is an algebraic group defined over k, the product morphism GxG .... G induces a group law on GA

k

which makes GA

a topolo-

k

gical locally compact group: the adele-group of G. If w is a gauge-form on G, ivariant by left-translations on G and if

(AV)

is a set of convergence factor for G, then the Tamaga-

wa measure rl = (w, (A»

is a 1eft Haar measure on GA. Thi s measure k

is independent of the choice of w (Theorem 2,3.1) and will be called

- 23 -

the Tamagawa measure for G derived from the convergence factors If (1) is a set of convergence factors, then

(w,(1»

(A V).

will be called

the Tamagawa measure for G. The group Gk is a discrete subgroup of GA

and we consider

k

the left homogeneous space GA IG k. We shall be interested in the follok

wing questions : (I)

~

GA IG k compact?

(II)

~

GA IG k of finite measure for any left Haar measure on

k

k

GA ? k

(III) If G is unimodular, if (II) is true, and if (1) a set of convergence factors for G, then what is the number (w,(1))? GA/G k

T(G)=!

k

Under the assumptions in (III), T(G)

will be called the Tama-

gawa number of G. For instance, if G= Gan (additive group in n variables), then (I) is true (a fortiori (II), and, by definition of' ]Jk' T(G) = 1If G is an algebraic group and w a left-invariant algebraic gauge-form,

w(xs)

is also left-invariant for all

of the form w(xs) = L'I(s)w(x)

SE

G, hence must be

where L'I is a character of G, called

the module of G. As in the topological case, w(x- 1) =L1(x- 1)w(x). If A=1, G is said to be unimodular (example: all reductive groups). If G is defined over k, then we may take w defined over ki therefore the module L'I also is defined over k and extends to a homomorphism L'lA : GA -r Ik = (Gm)A . Then IL'lA I where I I denotes the k k k k 1dele-module is a homomorphism of GA into which coincides ob-

R:

k

v10usly with the (topological) module of GA . Hence, if G is unimo~.

GA

is unimodular.

k

k

Let now H= Gig be an a I gebra i c homogeneous space of G. A

- 24 gauge-form w on H is called a relatively invariant form belonging to the character X of G if w(s·P)

=

X (s) ·w(P)

SE.G,Pe::.H

We have, as in the topological case, the theorem : Theorem 2.4.1. Denote by

and 0 the modules of G and g.

~

There exists a relatively invariant form on G/g belonging to the character X of G, if and only if the character o(o)~(o-1)

of g coin-

cides with X on g. Corollary. l! G and g are unimodular, there exists on G/g a gauge-form invariant by G. Let dx,

do,

dP be respectively a left-invariant gauge-form

on G, a left-invariant gauge-form on g, and a relatively invariant gauge-form on G/g

belonging to the character X of G; let

canonical mapping of G onto G/g, and put P = ~(x) ~*(dP)

is a differential form on G. Put a(x)

= xg

= ~*(dP),

~

be the

for XE. G. Then and let s(x)

be any differential form on G such that, for every Se: G, s(so) = do for

OE

g, i.e. such that s(sx)

induces on g the form do. It is

easily seen that the form a(x) II s(x)

on G is a gauge-form which does

not depend upon the choice of 13; this will be denoted symbolically by dp·do. We have dp·do =AX(x)dx, where A is a constant. If A= 1, we say that dx,

do,

Let G' dx',

do',

dP'

dP match together algebraicallay. be a locally compact group,

g'

a closed subgroup; if

are respectively a Haar measure on G', a Haar measure

on g', and a relatively invariant measure on G'/g' character X'

of G', we will say that dx',

do',

belonging to the dP'

topologically if the integral formula f

G'/g'

dP' ff(x'o')do' g'

=

ff(x')X'(x')dx' G'

match together

- 25 -

holds for every f in l 1 (G' ,dx'); in this formula, image x'g'

of x'

P'

means the

in G'/g'. This formula will be abbreviated symbo-

lically by dP"da' =X'(x')dx'. Theorem 2.4.2. Let G and gcG be algebraic groups defined .. over k. Suppose that the map G+G/g = H admits local cross-sections (in the sense of Theorem 1.2.2), and that the density condition of Theorem 1.2.2 is fulfilled. Then GA/9 A= HA• Straightforward application of Theorem 1.2.2. We note that the density condition of Theorem 1.2.2 is automatically verified if k is a number-field (Rosenlicht). For the function fields it is easily verified in each of the particular cases treated in Chapters III and IV. Theorem 2.4.3. Let G, g be as in Theorem 2.4.2. Let dx, da, dP be gauge-forms defined over k on G,g,G/g, matching together algebraically. Let (A V)'

(~v), (vv)

be three sets of factors for k with

AV = ~v 'vv' Then:

(1)

If two of three sets

(A V)'

(~v), (vv)

are sets of convergence

factors (for G, g, GIg respectively), so is the third one. (2)

.!!.

(1)

is satisfied, denoting dAx= (dx, (A V ))' dAa= (da,(u)),

dAP= (dP, (vv))' then dAx, dAa, dAP match together topologi~.

Moreover if dP·da=X(x)dx, then dAP'dAa=IXA(x)ldAx

where

I I is the idele-module and XA: GA+ Ik the adele extension of X: G+Gm.

The proof is straightforward and is left to the reader. Lemma 2.4.1. Let G be a locally compact unimodular group, I

g

unimodular subgroup of G, y a discrete subgroup of g. Let

dGX, dgU, dG/gx be Haar measures of G, g, GIg such that dGx. dG/gx.dgu. Let dg/yu and dGIl ~

be the measures induced by dgu

dGx in the local isomorphism g+g/y, G+G/y. Then we have

dG1y X. dG/gx'dg/yu

in the sense that

- 26 (1)

f f(X)d G/ X= f dG/gX f f(XU)d g/ u for G/y y G/g g/y y

f€L 1(G/y).

Slight modification of Fubini's theorem. Lemma 2.4.2. Let furthermore that

G:::>r::>y

and let

dG/rx

r

be a discrete subgroup such

be the Haar measure on

fl = f dg/ u~ +00. Then for 9 g/y y integrable continuous functions), we have

morphic to

dGx. Let

G/r

locally iso-

fE L+(G/g) (positive

(2)

where the two sides of (2) are both infinite or both finite and equal (as usual

00'0=0). f f(x)d G/ X= f dG/ x f f(xu)d / U=fl' f f(x).dG/gx; G/y y G/g 9 g/y 9 y 9 G/g

From (1)

on the other hand :

Lemma 2.4.3. With the same notations, suppose rflg=y. Put

H=G/g,

dGx = dHy·dgu, where p: G.... H. Then for

rH=rg/g~r/Y.

9 normal and

Assume furthermore that

YE. H is the image of

x EG

by the canonical map

fE.L+(H/r H), fl • f f(y)d H/ r y = f f(p(x))dG/rx. 9 H/r H H G/r

(3)

In particular, if flH = f dH/ r y, flG = f dG/rx, then: if two of the H/r H H G/r three numbers

flg'

flH' flG

are finite, so is the third one, and

flg 'flH = flG' By assumption we have

dHy= dG/gy· Then by formula (2),

- 27 -

But by Fubini,

If we put fry)

Z

=

h(yt;) , then we have

t;~fH

)J

f

g H/fH

f(y)d H/ f Y= f f(p(x) )dG/fx H G/f

But there are "sufficiently many" functions This proves (3). Putting f

=

Z h(yt;) t;f:f H

in L+(H/f H).

1 in (3), we get the second part.

Theorem 2.4.4. Let G and gcG be unimodular algebraic jroups defined over k; assume that g is normal; call

¢ the canoni-

cal mapping of G onto H= G/g. Put H' A= ¢A (G A), Hk =¢(G k); assume that HA is an open subgroup of HA; also assume that (1) is a set of COnvergence factors for g. Then every set

!2r

G is such a set for

(Av)

of convergence factors

H; and, if dAx, dAY are the corresponding

Tamagawa measures for GA, HA, we have, for f€ L+(H'A/Hk)

(4) Cdg)

=

Tamagawa number of g, if it is defined; otherwise +00).

As the kernel of the mapping ¢A of GA onto HA is gA' we Cln identify HA with GA/g A. As D~

Hip

HA is open in HA, the image of Gk v for every v, and the image of Go by ¢ is ~ is open in Hk -p v for almost all p. Let dx, dy, du be gauge-forms for G, Hand

- 28 -

g, matching together algebraically. By the theory of analytic groups over complete valued fields, there are, locally, analytic cross-sections for gk

in Gk

v

v

for each v; this is enough to guarantee (just as in

Theorem 2.4.3) that

(dx)v' (dy)v' (du)v match together topologically

for every v, with the same conclusion as in Theorem 2.4.3 about sets of convergence factors; in particular, if (1) is such a set for

g, the

sets of convergence factors for G and H are the same. Then dAx, dAY and the Tamagawa measure for g match together topologically on GA, HA and gA; (4) follows now from Lemma 2.4.3. Corollary. If at the same time G and g satisfy the conditions of Theorem 2.4.2 {existence of a cross-section and the density condition), then (4) holds with HA=H A, Hk=H k; and, if (1) is a set of convergence factors for G (or for H), ,(G) = ,(gh(H). Remark 1. If, instead of assuming that (1) is a set of convergence factors for g, we merely denote by

(A V)'

(vv)

(~v),

sets of

convergence factors for G, g and H satisfying AV = ~vvv for all

v

(as we may by Theorem 2.4.3), then the proof of Theorem 2.4.4 remains valid and shows that (4) holds when we take for dAx, dAY the Tamagawa measures for GA and HA corresponding to the sets ded we replace ,(g)

where dAu

(A V )' (vv)' provi-

by the "modified Tamagawa number"

is the Tamagawa measure for gA corresponding to the set

(~).

Remark 2. In the proofs of Theorems 2.4.2 (identification of HA with GA/g A) and 2.4.4 (identification of HA with GA/g A), we have made implicit use of the fact that, whenever a group such as GA acts on a locally compact space, then the mapping of GA onto any locally

- 29 -

closed orbit in that space is open, so that the orbit can be identified with the quotient of GA by the stability group of anyone of its points. This is a special case of a known theorem (cf. MontgomeryZippin, Top. Transf.-Groups, p. 65), proved by an elementary category argument. The same theorem will be used occasionally, sometimes without reference, in the next chapters.

- 30 -

CHAPTER III THE LINEAR, PROJECTIVE AND SYl"PLECTIC GROUPS 3.1. The zeta-function of a central division algebra As formerly, whenever V is a variety, defined over a field

k,

we denote by Vk the set of points of V, rational over k; a vectorspace of dimension dover k can always be denoted by Rk, where R is an affine space of dimension d in the sense of algebraic geometry. In particular, any algebra over k can be so written; the obvious extension to R of the multiplication-law on the algebra

Rk makes

R

into an algebra-variety, defined over k (which means that the multiplication-law on R is defined over k). The given algebra

Rk over k

is absolutely semisimple if and only if R is so, i.e. if and only if R is isomorphic (over the universal domain) to a direct sum of matrix algebras. Let Dk be a central division algebra of'dimension n2 over k. This algebra defines an algebra-variety D of center Z(Zk =k). Over

k, D is isomorphic to domain

Mn(~)

(total matrix algebra over the universal

~).

The reduced norm is a multiplicative mapping N: D+Z; it is a homogeneous polynomial function of degree n. Let D( 1)

be the subvariety of dimension n2 - 1 defined in

D by the equation N(x)

=

1. This is an algebraic group defined over k,

the special linear group of D.

- 31 -

In the projective space

~(D)

derived from D considered as

an affine space, the multiplication on D induces on the Zariski-open subset defined by the homogeneous relation N(x) f 0, a structure of algebraic group defined over k, the projective group of D. We can describe it alternatively in this manner: denote by D* the multiplicative (algebraic) group of the elements of D with non-zero norm. Then G= D*/Z* is isomorphic to the projective group of D. Furthermore, the restriction to D(1)

of the canonical mapping D*~G identifies D(1) /

(center of D(1)) with G. Let ai' 1;;; i ;;; n2 , be a bas i s of Dk over k, and 1et x =l:iaiXi

be a generic point of D. The set DQ =l:.a.O forms a ring 1 l-P p

for almost every p. Furtermore if p does not divide the discriminant of D then Do

is a maximal order of Dk ' and one has

-p

p

Dk = Mn(k p)' p

Since every maximal order of Mn(k p ) is conjugate to Mn (0-p ), there is II ring isomorphism Do ~ Mn(Q) if pf S, S being suitably chosen. -p

p

As an invariant measure on D*, we can take w = N(x )

-n dx 1... dx 2 n

and if pt. S, we have ].l

=/

w

=

p D* p GL Qp

x= (x 1·J·),

dX

=

/

(0 )

n -p

11 ..

1 ,J

IdetXI- n (dX) = / p P GL

(0 ) n -p

(dX)p

dx lJ .. , q = N(p) .

From this formula follows that the Ap = 1 - N(p)-1

are a set of

convergence factors for D*. The Tamagawa measure (w, (Ap)) will be cal1ed to

W,

•

A.

- 32 -

The mappi ng

N: D* -+ Z* "Gm can be extended to

N: DA -+ (Gm) A= I k• We denote by

I I the idele-module (2.1.1).

We take on DA the Tamagawa measure dX A = dx S x 11 pf.S dxp , _n 2 dx S = flK S dx . Then f dX A = 1• VEo V D /D

n

where

A k

Lemma 3.1.1 (Fujisaki) . the subset of Let

.!.!. m~

DA determined by

IN(x)1

the image in

~M

DA/Dk

of

is compact.

X be that subset. Take a compact subset C of DA whose

measure (for the measure

dx A) is> Mn and

C'=C+(-C)={cl-c2Icl,C2E"CL automorphisms of gular norm

O<m~M,

If

> m-n ; put

aE.D*A'

x-+ax

x-+xa

are

IN(a)l n (idele-module of the re-

DA whose module is

N(a)n); therefore, if ae X, the images

under the automorphisms

and

x-+a-\, x-+xa

of

DA

a- 1C, Ca

of

C

have a measure> 1 and

are not mapped in a one-to-one manner onto their images in DA/Dk by , the canonical homorphism of DA onto DA/D k, since the measure of DA/Dk

is

1. This is the same as to say that

are not empty, i.e. that there are c' =actEC', c"=sa-k C'. Then nite set

ct, SE Dk

c"c' =Sct

Sa = ~i; as

with its image in

DA x DA

rna.

which is a fi-

Dx D), DA

can be identified

under the corresponding adele-mapping; so the

of the point~ xl£D A such that

pact. We have thus proved that, whenever act

is in c,2nDk ,

Dx D under the mapping

x -+ (x,x- 1 ) (since that image is closed in

that

such that

c' = act, thi s gives

can be identified with its image in

Vi

C'a('lD k

{~l' ..• '~h} since Dk is discrete in DA and C,2 is com-

pact. Assume that

set

a- 1c'(,)D k and

(x,x-l)EC'X(~ilC') a~

is in the union of the compact sets

is com-

X, there is ctE Dk such Vi. This proves the lem-

- 33 Remark. Lemma 3.1.1 shows that Problem

(I)

of 2.4 ("is GA/G k

compact?") has an affirmative answer for the group D(1). Lemma 3.1.2. There is a constant c >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 O
A(1)=~.

For xE: DA, put

so that we have f + + f _ = 1; and write z!(s) = f f+(x) IN(x) IS(x)wA ' DA

Z~(s) = f f Jx) IN(x) Is(x)wA ' DA so that Z = Z! + z~. Clearly, if the integral for

z! converges absolu

tely for some s, it converges absolutely and uniformly for o

Re(s) < Re(s ); as the multiplicative calculation has shown that the in-

0

tegral for

Z, hence also the integral for

for Re(s) >n, 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

<1>;

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
where

fl(t)

f

fl( IN(x) I )wA

D*/D* A k

is the function defined on

6+

by

By lemma 3.1.2, this gives c ('¥(O) _
~('!'(O)

(c

s-n

L-.!..1_
qs-n_ 1

is the constant in Lemma 3.1.2). This shows that

meromorphic in the whole plane, with a residue at c'¥(O) (resp. c'¥(O)/log q). As we have

Z~ and Z

s = n equal to

'!'(O) = f D
with (3) gives

c = Pk (resp. c = Pk log q). This proves:

Theorem 3.1.1. (i) s

and of

Z (s)

is the sum of an entire function of

- 41 -

where Pk = [(s-1 )sk(sl] s=l' and '¥ is the Fourier transform of 4>. (il)

Z4>(s) = Z'¥(n-s).

(iii) We have the formula

provided F is such that the right-hand side is absolutely convergent

(N.B. In the function-field case, q is the number of elements of the field of constants of k). 3.2. The projective group of a central division algebra. Lemma 3.2.1. Let G be a locally compact unimodular group,

g

a closed subgroup of the center of G; put G' = G/g, and let dx, d'x', dgZ be Haar measures matching together topologically on G. G' , g. Let 6 be a discrete subgroup of G; put

H= G/6, 0 = I:,("\g,

6' =gMg = Mo, H' = G' /6 ' = G/g6, Y= g/o = gM6. Then y is a commutative group operating continuously on H; the quotient H/y is canonically isomorphic to .H'; and we have, for f€L+(H) f f (u) du = f diu 'ff ( tu) d t H H' y g ~

tu, for uEH, tE:Y, is the transform of u by t

II we have said, and

acting on H

u ' l£ the image yu of u in the canonical map-

21ng of H onto H' = H/y. The first assertion follows algebraically from the fact that g

11 in the center, and topologically from the fact that 6 is discrete in •• Now we have, under obvious assumptions on ¢, ¢', ¢" :

- 42 (x'=xg),

J 4>(x)dx = J d' x' J 4>(xZ)d gZ G

G'

9

J 4>'(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
= Jd'u' J ( L 4>(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
Dm over

k, it is the image M(x)e of e under the element

- 49 -

M(x) =

xl

0

x2

xl

x3

0

-1

0

0

0

0

0

0

'm-2 xm of G, where

'm-2

0

is the unit element of Rm- 2 . This is a "generic

cross-secti on" (i n the sense of Theorem 1.2.2) for the mappi ng X+ Xe of G into H= Gig; also,

Gk is obviously Zariski-dense in

G. We can

therefore apply Theorem 2.4.2 (cf. Remark 2 at the end of Chapter II), and therefore identify GA/9 A, and, for every v, Gk 19 k ' with the orbns v

HA, Hk

v

of e under GA and under Gk ' respectively; also, for alv v most every p, we can identify G Ig with the orbit HO of e Qp Qp -p under Go. For every v, there is an isomorphism p of Dk onto a v v -p matrix algebra M (D~) over a division algebra D~ with the center rv kyi this can be extended to an isomorphism which we also call Py ' of Rkv =Mm(Dk v) onto Mmrv(D~), and also to an isomorphism of (Ok)m onto the space of (mrv,r )-matrices over D~; as we have seen, v v Hk consists of the elements x E(Dk)m such that pv(x) has the rank y roy' For almost all

11 an

p,

v

we have rp = n,

m

x €(D o ) ,PpW -p (mn,n)-matrix over Qp ; this matrix, by reduction modulo p, de-

termines an (mn,n)-matrix pp(x) _. N(p)

D~

= kp' and, for

over the finite field

Fq with

elements; it is easily seen that we can choose S such that

this is so for p¢S, and also that, for P¢S, Ho

consists of the

-p

.1ements x of (Do)m such that pp(x) is of rank n. -p 2 The additive group D~ is isomorphic to (Ak)mn; it will now

Itt shown that following :

D~ - HA

is of measure

O. This is a special case of the

- 50 -

Lemma 3.4.1. Let w be a gauge-form on a non-singular variety V, defined over k; is a common set the measure measure sure

~

(AV)

(W,(A V))

(W,(A V))

V'

be a k-open subset of V; assume that there

of convergence factors for V and for for

V'. Then

VA is the measure induced on VA by the

for VA; and, for that measure,

VA - VA has the mea-

O. Let j

be the injection mapping of V'

into V; then

jA is

an injective mapping of VA into VA' which we use to identify VA with a subset of VA

(note that the topology of VA is in general not

that induced by VA' as shown by the case of Gm considered as the complement of {O} in Ga , which corresponds to the "natural" embedding of Ik

in Ak; in that case the two varieties have no common set of conver-

gence factors, and

I k, not Ak - I k, is of measure 0 in Ak). The

first assertion follows from the definition of V ,V Qp Qp I

(W,(A)). Now define

("almost intrinsically", cf. 2.2.3 and 2.3) by means of suita-

ble converings of V and V'; by Theorem 1.2.1, there is S such that V~

C Vo for p¢:S; let ]Jp,]Jp be the measures of -p -p "local measure" wp; by assumptions, np.A~1]Jp and

,V~

Vo -p

for the

n pAp-1 ]Jp

-P.

are absolutely convergent. For each v,put Xv=V k -Vk ;thisisan v

v

analytic subset of Vk and is therefore of measure 0 for the local v measure Wv (by the theory of analytic varieties over complete valued

Then, for the product-measure measure

TIp ( A~1 wp) on

PS"

PS' (S")

has the

- 51 -

Therefore, if we put QS' = US"PS'(S"), PS' -QS'

has the measure O.

We have thus shown that, in the product VS

the set of the poi nts

x = (x)

I

X

PS"

such that either Xv E: Xv for some v or

of measure

(X p)PES
0; since the complement of that set is contained in

VA'

this proves the lemma. In order to apply this to Dm and H, all we need show is that (1)

is a set of convergence factors for H=G/g, or (in view of Theorem

2.4.3) that this is so for

g. In fact,

g is the semi-direct product

of the groups consisting respectively of the matrices

( 1

0)

o X'

(X'EG ' ),

(01

and to Dm-1 ; our assertion follows now from Theorem 2.4.3 applied to g, Dm-1 and these groups being respectively isomorphic to G'

G' = g/D m- 1• Also, this shows that

g is unimodular; and, in view of the

fact that T(D m- 1) = 1 (since T(G a ) = 1), and that T(G ' ) = 1 by the induction assumption, Theorems 2.4.3 and 2.4.4 show that T(g) = 1. Now, applying Lemma 2.4.2 to the groups J f(x)dx =

HA

GA, gA' Gk, gk' we get

J

GA/G k

(

l:

f (XS) ) dX

~EHk

where dx and dX are Tamagawa measures in HA and GA respectively;

as DAm- HA is of measure 0, we may, in the left-hand side, replace the integral on HA by the integral on

D~, dx

being then (in view of Lem-

ma 3.4.1) the Tamagawa measure on D~. On the other hand, since Dk is • division algebra, we have Hk = D~ - {O}; therefore (1)

J f(x)dx =

Dm A

- 52 Here

f

may be taken as any function in D~ such that the left-hand

side is absolutely convergent. If at the same time we assume f

to be

such that f(Xx), for every XeGA' is of Poi sson type (one can select such a function by a procedure similar to that described in 3.1), the right-hand side can be transformed by Poisson's formula. Let g be the Fouri er transform of f:

(as before,

ty is the transpose of the column-vector y); replacing . h XEG , we see that the Fourier transform of x, Y by Xx, t X-1 y, Wlt A f(Xx) is g(tx-1y) since det X= 1 ; (1) gives now: -1

(~mg(tx 11) - f(O) )dX GA/G k ne Dk

J f(x)dx =

J

Dm A If in

(1)

we substitute

9

for f and t X-1

for X (which does not

change dX), and combine the two formulas, we get J [f{x) - g(x)]dx =

Dm A

J [g(O) - f(O)]dX GA/G k

Here the left-hand side has the value g(O) -f(O), and all integrals are absolutely convergent; choosing f so that g(O) >'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 = <±l iZi . This determines a homomorphism of the multiplicative group R* of the invertible elements of R into the corresponding group Z* for Z; we have Z* =niZi, with Zi= Rk./Pi*) ~Rk./k(Gm); this is a torus of dimension d=Lid i ; we 1

1·

(Z*\ ~ TIiki; and the mapping N induces on Z* an isogeny of

have

Z* onto itself, given by

In the special case treated in 3.3, the norm mapping N: z+zmn of

Z*

into itself was factored into z+zmn/v+zmn. Similarly, we as-

sume now that we have factored

N as follows:

where T is a commutative group-variety, defined over k, and

~,

v

Ire isogenies of Z* onto T and of Tonto Z*, also defined over k' then T is a torus, isogenous to Z*. We introduce the algebraic

groups : r== {(x,t)€R*xTlv(t) =N(x)} r o == {(z,ll(z)lzE:Z*} AI in 3.3,

fo

is the connected component of the identity in the center

- 56 -

of r; it is isomorphic to Z*; the group G= fir 0 R(1)

is isogenous to

(it is perhaps the most general group isogenous to R(1)

over k,

but this will not be discussed here), and we investigate ,(G). Call

<jJ the homomorphism of r

its kernel is the group r'

=

into T given by <jJ(x, t)

= t;

{(x,e)IXER(1)}, where e is the neutral

element in T, and is isomorphic to R(1). Lemma 3.3.2 can be generalized as follows Lemma 3.6.1.

<jJ(rA)/<jJ(r k) is canonically isomorphic to an open

subgroup of TA/Tk of finite index 2i. Just as in the proof of Lemma 3.3.2, one sees, by using Eichler's theorem, that c/l(rA)nT k is the same as <jJ(r k), and also that any t ETA is in <jJ(r A) whenever tvE: <jJ(r k ), i.e. whenever v v(tv)E: N(R k ), for every infinite place v of k. This proves the lemm~ v with i = 0, in the function-field case. In the number-field case, call S the set of the i nfi nite places of k; put Zs ZS,o

lTv,,-s Z* k ; call v the connected component of the identity in ZS' and put Z~ = Z*

x

11

Z* kp

s,o p¢S

as Zs

is a product of factors

=

~*, ~*, the group

te abelian group of type (2,2, ... ,2); call

0

y = Z;''/Z~ is a fi ni-

the canonical homomor-

phism of ZA onto y. From the remarks made above and in the proof of Lemma 3.3.2, it follows that, if t ETA is such that v(t) E: Z~, t

is

in c/l(r A); therefore <jJ(r A) contains the kernel of the homomorphism oov

of TA into y; as the index in our lemma is equal to the index

of <jJ(fA)T k in TA, this completes the proof.

(N.B. It has been shown by Serre and Tate that, if T is any torus over g (the rational numbers), and T~ o

is the connected compo-

nent of the identity in TB' then TB = T8'T g; from this one easily con-

- 57 -

cludes, firstly (using the operation Rk/Q ) that, if T is a torus over a number-field k, and T~ is defined from T as Z~ was defined above from Z*, TA = T~.\, and, secondly, that the index 2i

in Lemma

3.6.1 has always the value 1). Now we apply to the groups r, r', ro the method used in 3.3. For each

i, let vi

be the norm in

(Zi)k = ki over k (this is both

the reduced norm and the regular norm in the sense of the theory of algebras); this can be extended to a norm mapping vi

in Zi

(whi,ch is

a polynomial function of degree di = [k i : kJ), which induces on Zi a character, i.e. a representation of Zi into Gm, rational over k; moreover, the group of all such characters of Zi is generated by vi' and the group of all characters of Z* = TTiZi, rational over k, is generated by the characters

As U is an isogeny of Z*

into T, one concludes from this (by well-

known elementary arguments in the theory of algebraic toruses) that the group of the characters of T, rational over k, is generated by r characters Xi' and that the characters Xi

0

U of Z* generate a sub-

group of finite index of the similar group for Z*. In other words, if we write

a .. Xi(U(z)) = TTv.(z.) lJ , j

J

J

with integers a ij , we have det (aij);t O.

Now we need a Tamagawa measure for rA. As r

is isogenous to

R* • TT1Ri, we can use for r any set of convergence factors (>. ). V

Rt·

(TT1>. v (1) ) '

where

(>. (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
a maximal order of Rk , isomorphic to Mn(Qp) (this follows from the p

consideration of the discriminant). For every v, Rk (D~)

a matrix algebra Mm

is isomorphic to v

over a central division algebra

D~

over

v

kv; we ca 11

r~ the dimension of D~ over kv' so that n = mvr v. For

each P€S we select a maximal order Qp of Rk ; this is isomorphic p

to Mm (Qp)' where Qp is the maximal order of Dp. For every p, we p

denote by TI'

a prime element of Qp (a generator of the maximal two-

sided ideal in Qp); by the theory of p-adic algebras, this can be done r

in such a way that TI = TI' P is a prime element of Qp . The zeta-function of R is defined as (1)

where wA' as before, is the Tamagawa measure for R* with the convergence factors

1 - N(p)-1, and (x) = lTv
for x = (x) ERA' the


is any integer (for ptS, we

put a(p) = 0). For v = VA € So' we have kA = B or

~

(we write kA

instead of kv A' etc.), with kp=B for 1~p~r1' D~~B or ~ (~=qua­ ternion algebra over B), kl = ~ for r 1 + 1 ~ 1~ r 1 + r 2, Di ~~. Let Tr be the reduced trace in Rk over k, which we extend as usual to R, and to Rk

v

for each v. For each VAES o ' choose a positive involution

- 67 x->-x

in

for

RA, i.e. an involutory antiautomorphism such that

X;" 0; this can be done by transporting to

isomorphism of

the transpose, and in

~,the

where

AI..

~,the

is the identity in

quaternion conjugate in

is any constant

of the integers

(here

M

z->-z

>O.

~).

>0

RA, by means of some

mA(D~), the involution X->- ti(

RA with

Tr(xx)

tx

is

complex conjugate

Now we take

It will be seen that the choice of

a(p), and of the constants

AI..

does not affect

S, "the"

zeta-function except for an inessential factor. We have, by definition,

for

Xv = Zictiti' ti E:.kv' Just as in 3.1, we find

(2)

l( s)

= ~kn

2

lJ

-1

lv(s) , lv(S) = Av

1*

IN(x v ) I~¢v(xv)wv

kv In order to calculate tead of (3)

lp(S)

for a given

a(p), r p ' mp' Identifying lp(s)

= (1_q-1 )-1

Rk p with

f IN(x) Ipsw p , r.l

r.l

p
a, r, m ins-

Mm(k p ); we have

= M (0' )*()'IT' aM m p

Let U be the set of the invertible matrices in the ring the disjoint union of co sets trices as

in 3.1, except that

U''IT,aA, if one takes for 'IT'

(0')

m -p

Mm(Q~);r.l

is

A the same ma-

has now to be substituted for

'IT,

- 68 -

and that a ij

has now to run through a complete set of representatives d.

of Qp modulo rr' J. As wp tive group

is an invariant measure in the multiplica-

Rk , the integral in (3), taken over U'rr,aA, has the value p

IN(rr,aA)I~Juwp' Here

IN(rr,aA)l p is easily computed by remembering that

the regular norm in R is Qp; one finds

N(x)n, and that rr,r

IN(rr,aA)lp=q-N with

is a prime element of

N=Lidi +ma. Also, the number of

matrices A for given values of the di

is qM with M= rLi(i-1)di.

This gives

In order to compute the last integral, we change from the basis a basis

(6 i )

of Mm(Qp)

(a i ) to

considered as an Qp-module; for

x=Liaiti = LiSiui' Si = Ljnijaj' Ilijck p' we get: Jw

Up

If we put

= J(D dt.) U

i

1

P

d= det (Tr(aia j )), d'

= Idet(Il")I J(Tfdu.) 1J

=

Pu

i

1

P

det(Tr(SiSj))' where Tr is as be-

fore the reduced trace in Mm(Dp)

over kp' we have, by the theory of

the different in p-adic algebras,

Id' Ip = qn(m-n); since

d- 1d'

= det(llij)2,

Qp/rr'op

this gives the value of

Idet(llij)l p' Finally,

is the finite field with qr elements, and a,matrix in Mm(Qp)

is in U if and only if its reduction modulo rr'o' -p is a matrix of nonzero determinant over that field; the number of such matrices is

2 and JU(TTidui)p has the value q-rm \!. Thus:

Zp(s)

=

Idl-~ q-mas-n(n-m)/2(1_q-1)-1 p

TIi=o 1_{i-n 1_ q n-s

- 69 For pEtS, we have a = 0, m= n, r = 1. This shows that lTplp(S) so 1ute 1y convergent for

Re (s)

is ab-

>n.

Now we calculate the lA(s) : we identify RA with Mm(D A), for mA, rAj also, put p=[kA:~]' As

writing again m, r of measure

RA-Rt

is

0, we have

we recall that

Nand

kA, and, if z €k A,

Tr mean the reduced norm and trace in RA over Izl P if II is·-the "usual" absolute

IzlA means

value. Use the "Iwasawa decomposition" D= diag(ol, ••• ,om) and T=(t .. ) 1J

X=UDT, where

to' U=l m,

is the diagonal matrix with the elements

is triangular

Then, if dU, dD, dT denote

(t .. =O

for

1J

f(D)dUdDdT; here we can take for space of the vectors

i>j, t .. =1 for l
-

Haar measures on the groups

one sees at once that a ·Haar measure on dT

(tij)i<j' i.e.

-

{U}, {D}, {T}

RA must be of the form

the additive Haar measure in the dT= Tri<jd+t ij

if d+t

measure in the additive group D~; and we can take dD = Expressing now that f(D)dUdDdT

0i >0,

lTi (d%

is the i ).

is invariant under X-+-XDo' i.e. under

(U,D.T)-+-(U,DDo,D~lTDo)' we see that f, up to a constant factor, must be given by

f(D) =

TT 0~2p(m-2i+1) i

'

Also, we have

°

IN(X) IA = IN(D) IA = IT ~P 1

t

Tr( X'X) = r

m 2 E 0.(1 +

i=l

1

m E

j=i+1

t .. t .. ) 1J 1J

- 70 -

As wA is a Haar measure on

one finds now

R~,

immedia~ely

m-1 Z () C A- pns / 2 TT r(-zlrp(s-ri)) As = AA i =0 where CA is a constant factor, which can be computed by calculating ZA(n); this is easily done by changing the basis

(a i ) for a basis

adapted to the identification RA =Mm(D>.); one finds 1

_

n2/2

ZA(n) = Idl~2 (A/1Trp) p

In view of the product formula TTv1d1v = 1, the product for Z(s) not contain d, as was to be expected choice of the basis

(Z(s)

does

cannot depend upon the

(ai)).

Now we prove : Theorem 3.8.1. Let Rk be a central simple algebra over k. for every valuation

v of k, Rk

~

is isomorphic to a matrix algebra v

over kv' Rk

is isomorphic to a matrix algebra over k.

The calculation given above shows that, except for a factor of the form C1C~' with constant C1, C2, "the" zeta-function of R depends only upon the "ramification indices" "splits everywhere", i.e. if rv=1

rv; in particular, if R

for all

v, we have

s n-1 Z(s) = C1C2 TT F(s-i) ;=0

with r

r

F(s) = r(s/2) 1 r(s) 2 (;k(s) in particular, for n = 1, we see that F(s)

is "the" zeta-function of

k itself. In order to prove our theorem, it is clearly enough to show that a central division algebra over k, other than

k, cannot "split

- 71 everywhere", i.e, that this vision algebra unless

Z(s)

cannot be the zeta-function of a di-

n = 1. In fact, the additive calculation in 3.1

has shown that, for such an algebra, s=

° and

Z(s)

has no other poles than

s = n on the real axis; in particular,

s

=0,

s

= 1.

The above formula shows now that

s

=0,

s

=n,

and double poles at

s

= 1, ... ,n

Z(s)

F(s)

has the poles

has simple poles at

- 1; therefore

n = 1.

- 72 -

CHAPTER IV THE OTHER CLASS Ic.l\L GROUPS 4.1. Classification and general theorems. We consider only the algebraic groups, over a groundfield k, which, over the universal domain, are isogenous to products of simple groups of the three "classical" types: "special" 1inear, orthogonal and symplectic. Excluding the case of characteristic 2 (which has not been fully investigated) and certain "exceptional" forms of the orthogonal group in 8 variables (depending upon the principle of triality), such groups, up to isogeny, can be reduced to the following types, which will be called "classical" (the letter indicates the type over the universal ~omain,

and K denotes any separably algebraic extension of k) L1.

Special linear group (or projective group) over a division algebra

L2.

DK over K.

(a) Hermitian (i .e., "special" unitary) group for a hermitian form over a quadratic extension K' of K. (b) Id. for a non-commutative central division algebra DK, over K', with an involution inducing on K' non-trivial automorphism of K'

the

over K.

01.

Orthogonal group for a quadratic form over K.

02.

Antihermitian group for an anti hermitian (or "skewhermitian") form over a quaternion algebra over K, with its usual involution.

- 73 ,.

Sl.

Symplectic group over K.

S2.

Hermitian group for a hermitian form over a quaternion algebra over K, with its usual involution.

Our problem was to solve, if possible, the questions (I), (II), (III) listed in 2.4, for these various groups, their products and the groups isogenous to such products; only a fraction of this program will be fulfilled. In Chapter III, (I) has been solved (affirmatively) for 0(1), the special linear group of a division algebra (cf. Remark following Lemma 3.1.1); it will be shown presently that the answer to (1) is negative for

R(ll

when R=M (0), m>2; and the problem will be solved

m

-

for all remaining types except L2(b), for which only a partial answer will be given. Problem (II) has been solved affirmatively, in Chapter III for the types L1, Sl; an affirmative answer will be given to it, in thi s Chapter, for the types

L2 (a), 01, 52; perhaps the same method

could be applied also to the types

L2(b), 02, but this would require

computations which have not been carried out. Problem (III) has been

so~

ved, in Chapter III, for the types L1, Sl; it will be solved in this Chapter, for the types L2(a), 01, S2, by a method depending upon the construction of zeta-functions for quadratic forms (the same idea can be applied to the exceptional group G2 , as shown by Oemazure in the Appendix); there is at present no obvious way in which one could hope to ex-

tend this method to the types L2(b) and 02. In many special cases, once the problems 0), (II), (III) have been solved for a group G, it is not

too difficult to obtain a solution for a group G' isogenous to G; but general theorems by which this could be effected are still lacking.

Perhaps the method described in 3.6 could be generalized. From now on, we consider exclusively the types other than L1;

it 15 enough to consider the case K= k (since the case of an arbitrary

K

..

can be reduced to this by the operation RK/ k). The groups in questbn

- 74 can all be described as follows k' = k in the cases

Put k'

a quadratic extension of

such that

Ok'

01, 02, S1, S2; in the case L2, call

k. Let

0'

Ok"

0'; put let

x->-x

k'

over

k if

k'; let

~,

n2 be

0 = Rk , /k(O'), this being the same as

0' for

be an involution (i.e., an involutory anti au-

tomorphism) inducing the identity on phism of

be an algebra variety over

is a division algebra with the center

the dimension of k' = k. In

•

k'

f k;

k, and the non-trivial automor-

this can be extended in an obvious

manner to an involution in R'

=~m(O'),

0, defined over k. Take tR=Rk'/k(R') =Mm(O); then X->- X is an involution in

R,

defined over k; so is the mapping X->-S-1.tX· S if S is an element of t-S =:!: S. The "classical group" defined by these data is R'k such that the one given by (1) where

G= {XER*[tXSX=S, N(X)=1} N is the reduced norm, mapping

R*

into its center

Z*= Rk'/k(Gm). For the symplectic group (type S1), we have to take k' = k, n=1, ts = - S, and the relation

t xsx = S implies

shown by the consideration of the pfaffian of fore the same is true for the type

N(X) = 1 (as

S and of

t XSX ); there-

S2, since these two types are one

and the same over the universal domain. Similarly, in the case have

k' = k, n = 1, ts = S; and

tional condition

t X' S' X = S imp1 ies

N(X) =:!: 1 ; the addi-

N(X) = 1 serves to single out the component of the

identity in the group

t X· S· X= S; therefore the same holds for the type

02. On the other hand, in the case L2, the mapping group

01, we

t X' S· X= S onto the subgroup

U of

X->-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;

K
(y,z) + (z,y); if it maps have

S into

B = t A; it if maps a, a'

be vectors, other than 0, in mes is

let a

x+x

(A,B), where

into

be an isomorphism of of

DK

into

A, B are in

Mm(K), we

(b,c), (b' ,c'), b, c, b', c'

Km; and the assumption

F(a) = F(a')

must beco-

t c •A•b = tc'.A·b'; the statement (i) amounts to saying that there V in

Mm(K)*

such that

b' = Vb

and

DK

t A· c ' = tv- 1• t A•c ; (ii) says

- 79 -

that we can take Y such that det(Y) = '; these statements are easily verified. It seems to be an open question whether (ii) is still valid for the type L2(b). Combining the above lemmas with Theorem 2.4.2, we have now: Theorem 4.1.1. Let s be a vector other than 0 ~ O~; ~ g be the subgroup of G whi ch 1eaves s fi xed; put p = F( s), and 1et 2: be the sphere of radius

p in Om. Then (except for the case m= 2,

p = 0 and the case m= 1 of types 01, L2(a), and possibly for the type L2(b)), we have 2:* = Gig, 2:'K = GK/9 K for every field

K:::>k, 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 ~ <5 - 1, be a bas is of Dk over k,

Now 1et

wi th

£

o

=

1; ca 11

automorphism of the

Ev

the mappi ng x ->- x£v of Dm, cons i dered as an

(<5m)-dimensional affine space. Let L be the group

of all such automorphisms, which is isomorphic, over k, to M6m , the

full linear group in <5m variables; then R* = Mm{D)* is the subgroup Of the elements of L which commute with Ev for 1 -
•• (1.0, ... ,0)

of Dm under the group L; this is a Zariski-open sub-

'It of Dm. and the orbit H of e under R* is a Zariski-open subset

- 86 -

of Ho. Let Go be the group of type 01, consisting of the elements of L of determinant 1, leaving the quadratic form F invariant; the group G defined by 4.1 (1l is then a subgroup of G. As before, put L:* = L: n H;

L:* = L: nH

and put

o

0

; then

L:*

is a Zari ski -open subset of L:*o.

Assume, from now on, that om> 4; in view of Lemma 4.2.1, and of Lemma 3.4.1, the Tamagawa measure on

L:~,

derived from any gauge-forr.1 dw, in-

duces on L:* the Tamagawa measure derived from dw on L*, and L*o - L* is of measure O. Now take any 1;E:L"k g, 90

(F

is not of index 0), and call

the subgroups of G and Go 1eavi ng 1; fi xed; by Theorem 4.1.1,

we can identify form dw on

L~

L~

with Go/go' and L* with G/g. Choose the gauge-

so that it matches algebraically with invariant

gauge-forms on Go and go

(in the sense of 2.4); then the Tamagawa

measure dWA derived from it is invariant under

(Gol A, hence also

under GA; therefore it induces on LA the (uniquely determined) Tamagawa measure on LA which is invariant under GA. In order to complete the proof of Theorem 4.1.3, we have to show that LAne' te measure, for that measure, whenever C'

is compact in

now clear that it is enough to prove this for

L* o

is of finiD~

; it is

instead of L*; in

other words, it is enough to cons i der the quadrati c case (wi th m>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)dwA<+oo function

rnA

~ D~, defined for x = (xvlE::DA

by

where the ¢v are as follows: (i) for almost all

(x)

=

for every TTv¢v(xv)'

p, ¢p is the cha-

racteristic function of D~; (ii) for all p, ¢p is continuous with -p compact support; (iii) for any infinite place vA' 1<1\(x)I ~CAexp(-QA(x))' where CA is a constant and QA is a positive-definite quadratic form on Dm considered as a vector-space over 6. kA

- 87 -

We first show how this implies the finiteness of the measure of ZAnC' Dm

Qp

for compact C'. Take for
for every p; take for
Q\(x)<1, where Q\ is as in (iii); then cI>o(x) = TTv
is the

D~, so that

can be covered by finitely many translates of this neighborhood;

therefore the characteristic function of C'

is majorized by a finite

sum Z.cI> (x+a.), with a.€ DAm; as every term of that sum satisfies the 1 0 1 1 conditions for cI> in Theorem 4.2.1, this proves our assertion. Now we prove our theorem. It is clearly enough to consider the quadratic case (with m> 4); then Z* = Z - to}. Proceedi'ng as in similar calculations in Chapter III, we see that it is enough to show that all the factors in the product

are finite and that the product is absolutely convergent. As to the first point, let Q be the quadratic defined in the projective space of dimension m-

1

by the homogeneous equation F(x) = 0; let f be the

obvious mapping of Z* onto Q; for each v, this determines a mapping of Z* onto the compact space Qk ; and Zk is fibered over Qk kv v v v by that mapping, with the fibre k*. As Qk is compact, the finiteness v

v

of our integral will be proved provided we show that each point of Qk v has a neighborhood ~ such that the same integral, taken over f-1(~), is finite. Assume (as we may, since the characteristic is not 2) that coordinates have been taken so that F(x) = ziaix~; if ~

is a suitable

neighborhood of a point of Q where w, f 0, we can write, in f- 1(O).

w,

=t. Wi/W, =u i

for 2~i~m, so that zif,aiu~= -a,. It is

•• sily seen that the invariant gauge-form on

Z*

can be taken to De

- 88 -

dw = dw 2... dwm/a 1w1; this is equal to t m- 3w(u)dt. where w(u) gauge-form on

~.

This gives

Because of the conditions on

~v'

the last integral is absolutely con-

vergent and is a continuous function of u in Now take any p such that all the a i Qp and that ~p

is a

~,provided

m~ 3.

and 2 are units in

is the characteristic function of

(Qp)m; let TI be

a generator of the maximal ideal in Qp ' For every point w of ~k n(Qp)m, we can write

w=TIvW', where v~O, w' E:(Qp)m, w' $O(mod.p),

p

F(w') = 0; the latter conditions aumount to W' E

~~ ,

since ~* = ~ nH -p consists of the points x €(Qp)m

and we have seen in 3.4 that Ho -p such that x $ 0 (mod. p). As the formula given above for dw shows that the mapping w-+TIVW changes

(dw)p

into qv(2-m) (dw)p' with q = N(p),

this gives

(dw) -measure of ~~. By Lemma 4.2.1, TT ~ is abp -p pp solutely convergent (for m>4). 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
( x) d'

P P

x

P

is the characteristic function of D~; d'x -p

measure for

p

is the local

Vk , derived from the gauge-form dx 1... dx om for

V (the

p

xi

are the coordinates, for any basis of D~ over k) and from the

convergence factor

1 - q-1. Clearly this can also be \~ritten

(2)

where

(dx)p

is the additive measure in

(0 )om, normalized so that the

-p group has the measure 1; we may assume that the quadratic form F(x) has been written as

F(x) = L:iaix~. Then:

Lemma 4.3.2. For almost all

p, we have

Zp(s)= (1_q-s-m)(1_q-S-1)-1(1_q-2s-m(1 for

0=1, m~

Zp(s)= (1_EQ-m')(1_q-S-1)-1(1_Eq-s-m')-1 for om=2m' , where in the latter case E is

+1

or -1

according as

(-1) m' a,a 2... a om is or is not a guadratic residue in Qp molulo p. In fact, it will be shown that this is so whenever all 2 are units in Qp ' Let

TI

ai

and

be a prime element in Qp ; for x ~(Qp)om,

- 90 we can write

with

x='TT\\'

v~O, x'£(Qp)om, x' $0 mod. p, and get

with

Zp(s) = fIF(X)I~'(dx)p' the integral being taken over

x $ 0 mod. p. Now let (Qp)om mod. pV we know that

N, for v

v> 1, be the number of those solutions in -

for the congruence Nl

is

xdQp)om,

qom-1_ 1 if

F(x) $ 0 mod. pV om

which are $0 mod.p;

is odd, and

(qm'_c)(qm'-l+ c )

if

om = 2m'; and it is easily seen that set

N = qom-1 N for v> 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 : Z
(3)

Put Lemma 4.1.3,

r = Gx D*, and make it act on r2f.!

the subgroup of

Dm by

is the same as the orbit of r

leaving

sf.!

sf.!

((X,z) ,x) .... Xxz. By under

fixed; this consists of the

such that

XSf.! = sf.!z-1; when that is so, we have

therefore

ZZ = 1. In order to determine the structure of

coordinates so that

then, for

S,S

f.!

rA; call

r(f.!)

(X,z)

F(t;f.!) = F(sf.!z-1), and r(f.!), change

appear as matrices

(X,z)E. sf.!' X must be a matrix (f1

~,),

with

tx's'X' = S',

and, in the hermitian case, det(X') = z. This shows that, in tl-Je quaternionic case, of type r(\J)

r(f.!)

(S2)m_1

is isomorphic to

G'

x

belonging to the matrix

is isomorphic to the group

D(1), where

G'

is the group

S'; in the hermitian case,

G'* = {X' Itx,s,x, =S'} . Algebraically,

- 36 -

the orbit of sunder r is V, so that we can identify V, alge]1 braically (i.e., over the universal domain) with r/r(]1); moreover, it is easily seen that the gauge-form F(x)-om/2 dx on V is invariant under r, so that we can find invariant gauge-forms on rand r(]1) whi.ch match algebraically with that form on V. We now proceed with the proof in the quaternionic case, and will then indicate the changes required to adapt it to the hermitian case. By the induction assumption, (1) is a set of convergence factors for G

I,

for

and T(G

I )

= 1;

r(]1), and T(r(]1»

therefore (1) is a set of convergence factors =

1 (since T(D( 1»

2.4.3 and 2.4.4, we conclude, since the orbit is open in VA' and since V, that

(1_q-1)

(1-q

-1

reasoning as in Theorans

= 1);

rA ]1 of sunder ]1 ) is a set of convergence factors for ~

is a set of convergence factors for r

(and therefore,

since it is such a set for D*, that (1) is a set of convergence factors for G), and also that the Ta~agawa measure on rA' ri]1)

and VA' deri-

ved from matching gauge-forms and from these convergence factors, match together topologically when one identifies ~]1 with rA/ri]1). We can now apply Lemma 2.4.2 to rA' ri]1)'~]1 and to the discrete groups r k, r~]1); this gives

where the sum in the ri ght-hand side is extended to all

s 6r k/r~]1), i. e.

to the orbit of s]1 under r k; this is nothing else than the equivalence class of s]1

in Vk for the equivalence relation defined above;

therefore, when we take the sum of both sides over all the right-hand side gets extended to all

]1, the sum in

sE:V k. At the same time, we

- 97 have F(Xt;;z) = zF(t;;)z, with write

F(t;;)E. k*, hence

d' (X,z) = dX·d'z, where dX, d'z

IF(t;;) 1= 1; also, we can

are the Tamagawa measures, on

G and on D*, derived from the convergence factors (1) and

(1_q-1),

respectively. Therefore:

(4)

and the multiplicative calculation shows that this is absolutely convergent for

Re(s)

>O.

this into two parts, gral a factor lizl

is

Just as in the similar calculation in 3.1, we spl it z!(s)

z~(s), by introducing into the inte-,

and

1 =f + (z) +f - (z), where

<1, = 1

or

>1,

and

f (z) -

=f

f + is +

0,

l

~

(z-,); then

or

1 according as

ill(s) +

is an entire

function. In z~(s), we apply Poisson summation, observing that Vk = Dkm- Ek, where E is the sphere of radius F(x) = 0 in

Dm. If IjI is the Fourier transform of
transform of
0, i.e. the variety

lizl-o m/ 21j1(X'yi- 1) with E
t;;~D~ On the other hand,

of xoSD~ for a given

X' = S-l.t X-l· S; this gives

1

=

Izzl-2o m E IjI(X'ni- 1)

n~D~ (X,z) + (X' ,i- 1 ) is a measure-preserving automor-

phism of rA' which maps

r k onto itself; if we make that substitution

in the integral for z!(-s-~om), and compare it with the expression for Z~(s) obtained by Poisson summation as we have just said, we get:

If

F is of index

0, then

Ek ={O}, and the calculation can be comple-

- 98 -

ted immediately by applying Theorem 3.1.1 (iii) to D*; it shows that ,(G) for

is finite, since the right-hand side must be absolutely convergent Re(s)

>0,

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 z<5m/2 = 1. We have found it more convenient to use f, rather tive; in fact,

(X,z)

than the effective group which could be derived from it. On the other hand, for even m>4

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 2Z
is an entire function and consequently

+

11

as

Io is a meromorphic

function, and that both of them satisfy "functional equations" similar to (5). The method fails for odd m. 4.5. The Tamagawa number of the orthogonal group. From now on, we consider exclusively the quadratic case <5 = 1. Our purpose is to prove, by induction on m, the Siegel-Tamagawa theorem T(G) = 2 (this can be shown, by purely formal calculations, to be equivalent to Siegel's main theorem on the number of representations of a quadratic form by a genus of quadratic forms). The reduction from m to m-l

can be effected, for even m, by the consideration of the fol-

lowing zeta-function

- 101 where V, ¢, d'x are as explained in 4.4, and VA+ is defined, as in 4.4, by \(F(x)) = 1, except that here dratic extension

k'

\

is the character of the qua-

k' = k(6 1/ 2), 6 = (_1)m/2 det(S). The

of k given

method used in 4.4 can be applied with small changes. This fails, however, for odd m (because the group of similarity transformations has not the same structure for odd m as for even m). The following treatment is valid for all values of m. We change our notation by writing F for the matrix S, so that we have now F(x) = t xFx . Whenever convenient, we may assume that F has been put into diagonal form,

F(x) = Eiaix~. For m even, we denote

by 6 the discriminant of F, 6= (_1)m/2 det(F); for m odd, we write 0= (_1)(m-1)/2 det(F). By Hasse's theorem, if p€.k*, and E is the sphere of radius p, Ek

is empty if and only if EA is empty, or also if and only if

there is

v such that Ek

is empty. v

Lemma 4.5.1. If P€k* and E is the sphere of radius

p, and

if EA is not empty, (1) is a set of convergence factors for E, provided m>4. From the formula (cf. proof of Lemma 4.2.1) for the number of solutions of a homogeneous quadratic equation mod. p, one deduces the number of solutions of F(x):: p mod. p in the field

Fq with q = N(p)

elements; assuming p to be such that FEO: Mm(Qp)' p E:Qp' and that 2p det(F) with

e;

is a unit in Qp' this number is qm'-l (q m' - d for m= 2m',

= (Mp) (quadratic residue character of the discriminant 6 mo-

dulo p); it is

qm'(qm' +n)

if m=2m'+1, with n= (Dp/p),

0- (_l)m' det(F). The conclusion follows now from Theorem 2.2.5.

For pE:k*, we consider the variety F(x)=pl

in the affine

space of dimension m+l, and, on that variety, the Zariski-open subset T(p)

defined by y +0; this is isomorphic to Ex Gm, if E is the

- 102 sphere of radius !:xGm

p (the mapping

(x,y)

+

(xy,y)

onto T(p)); in view of Lemma 4.5.1,

is an isomorphism of

(1_q-l)

is therefore a set

of convergence factors for T(p), for m~ 4. As dx = dx 1dx 2... dX m is a gauge-form on the (non-singular) variety T(p), we can take on T(p) the Tamagawa measure d'x derived from this and the factors we introduce, for each

(1_q-1).

pc. k*, the "spherical zeta-function"

where ¢ is a function in Am k, of the type described in 4.4. It should be understood that this is

0 if l:A' and consequently T(P)A' are emp-

ty. For every Ac.k*, we have Z(S,PA 2) = Z(s,p), as we see by making the change of variables

1

,.

(x,y) + (X,A - y), which maps T(p)

2

onto T(pA)

and leaves the integrand in Z(s,p) invariant. The zeta-function for the quadratic form F will be defined as (2)

Z(s) =

!:

p€k*/k*

2 Z(s,p) ,

where the sum is taken over a full set of representatives of k* modulo (k*)2. As usual, we start with a multiplicative calculation (which will give the proof for convergence, and the principal residue), and this depends upon the calculation of the local zeta-functions for almost all p. Take a finite set S of valuations of k, containing as usual all the infinite places, and such that, for piS: (il all coefficients of F are in Qp; (ii) 2det(F)

is a unit in Qp; (iii) the factor

occurring in the definition of ¢ is the characteristic function of (o)m (there will be further conditions on S later on). For any -p u E: kp' cons i der the integra 1 (3)

~p

- 103 -

As before, we see that fore, if we write

zp(s,ui) =zp(s,u)

u = T/J.V , where

v is a unit in

vck

p; there-

Qp' Zp(s,u)

can de-

for

pend only upon amod. 2 and upon the quadratic residue character of v mod. p (in fact, since which is

==

-p every element of 0p (k p)2). The value of Zp(s,u) is given by

1 mod. p is in

Lemma 4.5.2. Put

2 is a unit in

(vip)

u = 7Ta V, where

0,

v is a unit in

Qp • Then:

(il for odd m, m=2m' +1, we have

( 1-q -2s-2) ( 1-q -2s-m) Zp ( s,u ) =

where

~

q-s-1 ( 1-q 1-m)

for

( 1+nq -m')( 1-nq -2s-2-m')

for a even,

a odd,

n= (Dv/p), D= (_1)m' det(F). (i i) for even m, m= 2m', we have for a odd

for a where

E

= (ll/p), 6. = (-1) m' det(F). Assume that we have put

F into diagonal form,

We have to take the integral (1) over the set of points for'which

>.

2 F(x) = l:iaixi.

(x,y)

of

k~+1

F(x) =u/, XE(Qp)m, YfO; for each such point, we can write,

in one and only one way, where

even,

-1

Y x and y

ZE(Qp)m, tcQp' z$O mod. p and

-1

A

11

in the form y X=TI z, y=TIl-'t,

qo

mod. p; then we have

+]J~ 0 (since x must be in (Qp)m), and F(z) = TIa - 2A V and therefore

2>..s. a. We spl it up our domain of integration into the open subsets such that, on each of these sets, yen values

have given values, and

z,t

have gi-

z,t modulo p; the latter must then be such that

F(z) :;: v mod. p if

IIsume that

A,lJ

A,]J,Z,t

2A = a, and

F(z):: 0 mod. p if

are so given. and e.g.

z1

2A
*0 mod.

p; then, for

- 104 every choice of Z2"",zm such that zi=zi mod. p for 2~i~m, the equation F(z) ~ na-2\ v has exactly one solution z1 such that Z1 = z1 mod. p, and this is an analytic function of z2'" :'zm; therefore, on the subset of the domain of integration determined by those values of \,~,z,t,

we can use z2"",zm' t

(x,y)+(z2"",zm,t) subset of

as local coordinates, and the mapping

is an analytic homeomorphism of that set onto the

( Qp)m given by

zi = zi' t= t mod. p. On that set, and in

terms of those local coordinates, the gauge-form dx 1... dx m is given by

which gives, on that set, since a1z1 on that set is a unit in Qp

therefore, on that set, our integral has the value qN, with N ~ (s+1 )(2\-a) - m(\+~+1) .

For given values of

\,~,

this gives to our integral a contribution

(q-1 )v(v)qN if 2A ~ a, where v(v) F(z) = v mod. p, and

(q-1 )v(O)qN

is the number of solutions of

if 2A
is the number

of solutions, other than 0, of F(z) =0 mod. p; we have to take the sum of these contributions over all values of \,

~

satisfying

2A ~ a, \ + ~ ~ O. Us i ng the formul as gi ven above (i n the proofs of Lemmas 4.2.1 and 4.5.1) for v(v), v(O), we get our result. Now, for p If. S, denote by Up the multiplicative group of the units in Qp' and put HS group of I k' Also, write

=

(lk)2TTptsUp; clearly HS

is an open sub-

ISI for the number of elements of S.

Lemma 4.5.3. The group HSk* is of finite index in ver, if S is large enough, this index is

lk' Moreo-

[lk: HSk*] ~ 2 1SI , and we ha-

- 105 -

ve k*n HS

=

k*2

The characters of Ik which are 1 on

Ik/HSk* are those characters of order 2 of

k*, and on Up for every p E:S; by class-field

theory, they are the characters attached to those quadractic extensions 1

k(d 2) of k which are unramified outside S. In the number-field case, take S such that it contains all the primes dividing 2, and that the 1

primes in S generate the group C of ideal-classes of k. If k(d 2 ) is unramified outside S, every prime pti;S must occur in d with an even exponent, so that we can write the principal ideal

(d)

duct of primes PES and of the square m2 of an ideal sumpti on on S, we can wri te m as

Am'

with

as a pro-

m; by our as-

AE k*, where m'

has no

prime factor poi: S; therefore every quadratic extens i on of k, unrami:!.

fied outside S, can be written as

k(d 2 ) with dECE S' where ES is

the group of the S-units of k (elements of k which are p-units for every ptS). The index we have to compute is then known (Dirichlet-Cheval ley) that ES

[ES: E~. It is well-

is the product of a finite cyclic

group of even order, and of a free abelian group with

lsi - 1

genera-

tors. The proof is even simpler in the function-field case. This proves the first part (note that, if the index is finite for large S, it must 1

be finite for all

S). Now take dE:k*nH S; then, in

k'

= k(d i ),

each

P€S splits (i.e. can be extended to two distinct valuations of k') and each p ItS

is unramified. As there are only finitely many non1

ramified quadratic extensions of k, this implies d2 €k* provided S has been so chosen that, to every non-ramified quadratic extension of k, there is a least one pES which does not split in

k'

k'.

Now take any S satisfying the conditions in Lemma 4.5.3 as well as the earlier ones. Also, let V be, as in 4.3, the open set F(x)

+0 in the affine m-space; and call S"lS the open subset of VA de-

termined by the condition F(x)cHSk*; if y(S)

is the group of order

- 106 -

2 1S1

consisting of the characters of Ik which are

can a1so be defi ned as the subset of VA where

1 on HSk*, ~S

A(F (x)) = 1 for every

AEy(S}. Now we introduce the function ZS(s}= JIF(x}ls(x}d'x= 2- IS1 ~S

l: JIF(x}lsA(F(x}}(x}d'x AEY(S} VA

The multiplicative calculation for the 2 1S1 is the same as the one given in 4.4 for

integrals in the last sum

10 , 11

(this depends only upon

Lemma 4.3.2); it shows that these integrals are absolutely convergent for Re(s}

>0, 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) <1 + q-m

I

- 108 As m> 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

<11' (S) for Re(s) >0;

is absolutely'convergent,

remains between the same constants. In view

of (4), this shows that lim. info sZ(s) are between Il(S)

(1 +q -m' ) ,

p~

and lim. sup. sZ(s), for s =0,

and 1l'(S); since we may here take S as large as we

please, we have proved (8)

so that this formula holds for all

m> 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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[2]

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~]

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