THE SPACE OF O-ADIC NORMS

THE SPACE OF O-ADIC NORMS

BY

O. G O L D M A N a n d N . I W A H O R I Institute for Advanced Study, Princeton, N . J . , U. S. A.(1)

I n t r o d u c t i o n

The s y m m e t r i c space associated to t h e o r t h o g o n a l g r o u p of a real indefinite quad- r a t i c f o r m ~ c a n be described, as is well known, as t h e set M of positive definite q u a d r a t i c forms y~ which are minimal with respect to t h e p r o p e r t y I~l ~< YJ. T h e or- t h o g o n a l g r o u p 0 ( ~ ) of t h e f o r m ~ acts t r a n s i t i v e l y on M, a n d t h e isotropy g r o u p o f a n y ~ E M is a m a x i m a l c o m p a c t s u b g r o u p of 0(~). A similar s t a t e m e n t holds for

t h e symplectic group.

A. Weil raised t h e question of t h e p-adic analogue of this p h e n o m e n o n , a n d

~suggested the use of norms (sec. 1) in place of t h e positive definite quadratic forms.

If ~ is a non-degenerate q u a d r a t i c f o r m on a v e c t o r space E over a p-adic field we associate t o T t h e set ~ ( ~ ) (see sec. 4) of norms ~ on E which are minimal with respect t o t h e p r o p e r t y I ~ 1 ~ 2. T h e n again t h e o r t h o g o n a l group O ( T ) o f ~ acts t r a n s i t i v e l y on ~(~0). However, t h e isotropy group of a n element a E~(~0), while still c o m p a c t , is no longer a m a x i m a l c o m p a c t subgroup.

T h e s t u d y of n o r m s on p-adic vector spaces it n o t new. (See for example Cohen [1]

a n d Monna [3].) These a u t h o r s were concerned with t h e metric t o p o l o g y induced on t h e v e c t o r space b y a n o r m on t h a t space. W e are here concerned with t h e intrinsic

~structure t h a t is carried b y the set ~ ( E ) of all norms on a given vector space E.

W e define a n a t u r a l metric on ~ ( E ) , a n d in sec. 2 of t h e present p a p e r describe some o f the properties of ~ ( E ) as a metric space. F o r example, ~ ( E ) i s a complete, locally c o m p a c t arc-wise connected space, a n d is even contractible to a point.

(1) This research was partially supported by the National Science Foundation through Brandeis University, the University of California at Berkeley, Harvard University and the Institute for Ad- vanced Study.

10 - 632918. Acta mathematica 109. Imprira6 le 13 juin 1963.

138 O. G O L D M A N A N D N . I W k H O R I

The group A u t (E) of linear automorphisms of E acts, in a natural way, on ~ ( E ) leaving the metric invariant. I n sec. 3 we give some of the properties of this action.

The isotropy group of a n y point of ~(E) is a compact open subgroup of A u t (E), while:

the orbit of a n y point is a closed discrete subspace. The quotient space Aut

(E)\~I(E).

is proved to be canonically homeomorphic to the symmetric product of n circles, where n is the dimension of E. I n the same section we describe invariants for the:

conjugacy of two elements of ~(E) with respect to the isotropy group of a preas- signed norm.

I n sec. 4 we consider some relations between quadratic forms and norms a n d give a characterization of the elements of ~ ( T ) . I n this we make use of the relations, developed in sec. l, between norms and lattices in E. Using this criterion of mini- mality, we obtain an explicit description of the elements of ~ ( ~ ) from which the.

transitivity of O(q0) on ~(q0) follovr easily. To some extent, our technique in this section is a variation of the method used b y Eichler [2] in his study of the action of O(~0) on certain lattices in E associated to ~.

I n the fifth section we introduce the notion of the discriminant of a quadratic form with respect to a norm and describe some of the properties and applications of this concept.

We have given such a detailed description of ~(E) as a metric space and of t h e action of Aut (E) on ~/(E) because we feel that there are possibilities of using these structures to study other types of arithmetic problems which m a y be formulated in the p-adic domain.

We should like to t h a n k A. Wei| for suggesting this line of investigation a n d for his help during the course of our work.

Section 1. Lattices and Norms

The following notation will have a fixed meaning throughout this paper:

K is a field complete in a discrete valuation, having a finite residue class field, hence locally compact.

q is the number of elements in t h a t residue class field.

[I is the valuation of K normalized so as to assume all integral powers of q on g * .

is the valuation ring of K.

p = ~ e ~ is the maximal ideal of ~ .

THE SPACE OF ~)oADIC NORMS 1 3 9

E is a v e c t o r space o v e r K of f i n i t e d i m e n s i o n n, t o p o l o g i z e d in t h e o n l y n a t u r a l t o p o l o g y o v e r K , so t h a t E is l o c a l l y c o m p a c t .

P(E) is t h e p r o j e c t i v e space of E, so t h a t P(E) is c o m p a c t .

B y a lattice in E will be m e a n t a f i n i t e l y g e n e r a t e d ~ - s u b m o d u l e L of E w h i c h s p a n s E o v e r K . A l t e r n a t i v e l y , L m a y be d e s c r i b e d as a n ~ - s u b m o d u l e of E w h i c h is c o m p a c t a n d open.

L e t L be a l a t t i c e on E . I f x is a n y n o n - z e r o e l e m e n t of E, t h e set of e l e m e n t s a E K such t h a t a x E L is a f r a c t i o n a l i d e a l of 9 , a n d so h a s the" f o r m pro. W e set a(x) = qm. W e c o m p l e t e t h e d e f i n i t i o n of ~ b y s e t t i n g a ( 0 ) = 0 . T h e n , i t is r e a d i l y v e r i f i e d t h a t ~ h a s t h e following p r o p e r t i e s :

1. if x # 0 , t h e n a ( x ) > 0 . 2. ~(ax)= ]a] ~(x), a E K . 3. ~ ( x + y) ~< sup (a(x), a(y)).

A n y r e a l - v a l u e d f u n c t i o n on E h a v i n g t h e s e t h r e e p r o p e r t i e s will be called a norm on E. I n p a r t i c u l a r , e v e r y l a t t i c e d e t e r m i n e s a n o r m , a n d conversely, if a is t h e n o r m a s s o c i a t e d t o a l a t t i c e L, t h e n L is in t u r n d e t e r m i n e d b y a t h r o u g h t h e r e l a t i o n L={xEEIo~(x)<~l }. I t will be seen s h o r t l y t h a t n o t e v e r y n o r m is d e t e r m i n e d b y a l a t t i c e . W e d e n o t e b y ~/, o r b y ~ ( E ) , t h e set of all n o r m s on E , a n d b y s t h e s e t of n o r m s d e t e r m i n e d b y t h e l a t t i c e s of E .

I t follows i m m e d i a t e l y f r o m t h e d e f i n i t i o n t h a t e a c h n o r m is a c o n t i n u o u s func- t i o n on E . I f a n o r m is m u l t i p l i e d b y a p o s i t i v e r e a l n u m b e r , t h e r e s u l t is a g a i n a n o r m . F i n a l l y , 7 / i s p a r t i a l l y o r d e r e d b y t h e r e l a t i o n ~ ~< fl m e a n s ~(x) ~< fl(x) for all x E E . L e t L be a l a t t i c e . As ~ is a p r i n c i p a l i d e a l ring, L is a free D - m o d u l e ; l e t xl . . . . , xn be a set of free g e n e r a t o r s of L. I f ~ is t h e n o r m a s s o c i a t e d t o L, i t is c l e a r t h a t sr a,x~) = sup (I a, I).

P R O P O S I T I O N 1.1. Let ~ be any norm on E. Then there is a basis {x~} o] E, and positive real numbers r~ such that

a ( 5 aixi) = sup (ri [as 1).

Proo/. W e p r o v e t h e p r o p o s i t i o n b y i n d u c t i o n on t h e d i m e n s i o n of E . I f d i m E = 1, l e t x be a n y n o n - z e r o e l e m e n t of E . Then, a(ax) = a(x) [ a [ , a n d t h e a s s e r t i o n is v e r i f i e d in t h a t case. A s s u m e t h a t d i m E = n, a n d t h a t t h e p r o p o s i t i o n is v a l i d for spaces of d i m e n s i o n less t h a n n. L e t E 1 be a s u b s p a e e of d i m e n s i o n n - 1 of E . I t is clear t h a t t h e r e s t r i c t i o n of sr t o E 1 lies in ~/(EI) , so t h a t sr r e s t r i c t e d to E 1 h a s t h e f o r m de- s c r i b e d in t h e p r o p o s i t i o n .

140 O. G O L D M A N A N D N . I W A H O R I

L e t ~t be a l i n e a r f u n c t i o n a l o n E h a v i n g E 1 for its k e r n e l . W i t h x a n y n o n - zero e l e m e n t of E , c o n s i d e r t h e q u o t i e n t ]~t(x)I/a(x). F i r s t of a l l t h i s q u o t i e n t is con- t i n u o u s . S e c o n d l y , i t is a h o m o g e n e o u s f u n c t i o n of d e g r e e 0. H e n c e i t defines a con- t i n u o u s f u n c t i o n on t h e p r o j e c t i v e space P(E). As P(E) is c o m p a c t , t h i s f u n c t i o n a s s u m e s its m a x i m u m . Thus, t h e r e is a n e l e m e n t x~ E E such t h a t

I~(~)l< I~(~1)1

a(x) a(xl) '

I t is clear t h a t x 1 c a n n o t be in E 1. H e n c e , E is t h e d i r e c t s u m of K x 1 a n d E 1.

L e t y E E ; t h e n y=xl2(y)/X(Xl)+Z, w i t h z E E t. W e h a v e , on t h e one h a n d ,

/l~(y)l )

~(y)~<sup [ ~ ( x l ) , a(z) ,

a n d , on t h e o t h e r h a n d , f r o m t h e d e f i n i t i o n of Xl, t h a t

/> I~(y)l ~(~1)

I t follows t h a t

~(y) = sup ~ /I (y)I ~(~1), ~(z)).

F r o m t h e i n d u c t i v e a s s u m p t i o n , t h e r e is a basis x~ ... xn of E 1 a n d p o s i t i v e r e a l n u m b e r s r e . . . rn such t h a t

o~ a,x, = s u p (% [az[ ... rn lanl).

I f we set r 1 = a(xl), t h e n t h e a r g u m e n t a b o v e shows t h a t a ( ~ a,x,) = sup (r, ]a,I).

I f a e ~ ( E ) , a n d {x,} is a basis such t h a t a ( b a , x,)=sup(r, la, l), t h e n we s h a l l s a y t h a t a is canonical w i t h r e s p e c t t o {x,}.

W e d e n o t e b y E* t h e d u a l space of E . W e shall n o w d e s c r i b e a useful m a p p i n g f r o m ~ ( E ) t o ~ ( E * ) . L e t a e ~ ( E ) . I f • e E * , we h a v e a l r e a d y c o n s i d e r e d in t h e p r o o f a b o v e , t h e q u o t i e n t 12(x)l/a(x) for n o n - z e r o x e E . T h e c o n t i n u i t y of t h i s q u o t i e n t a n d "the c o m p a c t n e s s of P(E) e n a b l e us t o set

a*(Z) = sup !Z(x) l a(x)

T h e r e is no d i f f i c u l t y in v e r i f y i n g t h a t a* is a n o r m on E*.

T H E S P A C E OF p-ADIC NORMS

P R O P O S I T I O N 1.2. The mapping o~-->o~* has the /ollowinq properties:

1.

2.

3.

4.

141

(aa)* = 1/ac~* /or a > O.

i/ ~ <~ 8, then 8" <~ ~*.

i/ o~ is canonical with respect to a basis {x,}, and {~i} is the dual basis o/ {x~}, then c~* is canonical with respect to { ~ } and o~*(,~i)= o~(xi) -1.

Proo/. These a s s e r t i o n s a r e all t r i v i a l consequences of t h e d e f i n i t i o n of a*. T h e f o u r t h s t a t e m e n t follows f r o m t h e t h i r d b y m a k i n g use of P r o p . 1.1.

P R O P O S I T I O N 1.3.(1) Let ~ and 8 be any two norms. Then there is a basis o/ E with respect to which both ~r and fl are canonical.

Proo/. T h e p r o o f is b y i n d u c t i o n on d i m E ; t h e case w h e r e d i m E = 1 is t r i v i a l . S u p p o s e t h e a s s e r t i o n v a l i d for spaces of d i m e n s i o n less t h a n n. W e consider the"

q u o t i e n t ~ ( x ) / 8 ( x ) for n o n - z e r o x EE. T h i s defines a c o n t i n u o u s f u n c t i o n on P{E), hence a t t a i n s its m a x i m u m s o m e w h e r e . T h u s t h e r e is a n o n - z e r o x 1 E E such t h a t

6~(X) < 6~(Xl)

8(x) ~ all ~eE. (1)

F r o m t h e f a c t t h a t ~ = ~**, we h a v e

~(xl) = sup~ ~ ,

a n d as P(E*) is c o m p a c t , t h e

~1E E*, w i t h

s u p r e m u m is a t t a i n e d somewhere. Thus, t h e r e is a

, ( ~ ) . (2)

Clearly ~l(Xl)::~0. U s i n g t h e d e f i n i t i o n of a*, we h a v e f r o m (2)

IXI(x)] ~< a(x) (3)

Since f r o m (1) we h a v e 8(x)/8(Xl)>~a(x)/~(xl) , i t follows t h a t

I~l(x)[~ fl(x)

~ 8 ( x l ) " (4)

(1) This theorem was given by A. Weil in a course in algebraic number theory at Princeton University.

142 O. GOLDMAI~I A N D N . I W A H O R I

L e t E 1 be the kernel of

~1"

~l(Zl):~0,

F r o m the inductive a s s u m p t i o n , t h e r e is a basis {x~ . . . x,} of E 1 w i t h r e s p e c t t, which ~ ]E, a n d fl I~, are canonical. I t follows i m m e d i a t e l y f r o m (3) a n d (4) t h a t a a n d / are canonical w i t h respect t o the basis {x I . . . x,}.

COROLLARY 1.4. Let L and L' be lattices in E. Then there is a set o/ /ree gen erators (x 1 .. . . . xn} of L such that /or suitable aiEK*, {azx 1 .. . . . anx,} is a set o/ /re, generators o / L ' .

Proo]. L e t ~, ~' b e respectively t h e n o r m s of t h e lattices L, L'. B y Proposition 1.3 t h e r e is a basis Yl . . . . , y , of E w i t h respect t o which ~ a n d ~' are b o t h canonical Since the values a s s u m e d b y g a n d ~' are integral powers of q, t h e r e are element~, b~EK* such t h a t ~(y~)=lb~[. Set x~=b~Xy~. T h e n it is clear t h a t {Xl . . . xn} is a set of free generators for L. I f we choose a s E K* such t h a t ~' (xi) = ]a~

I -x,

will be a set of free g e n e r a t o r s of L'.

L e t {aj; ]" E J } be a n indexed f a m i l y of e l e m e n t s of ~ which has t h e following p r o p e r t y : for each x E E , the set {~j(x)) of real n u m b e r s is b o u n d e d . I f we t h e n p u t fl(x)=supj~j{~j(x)}, t h e n there is no difficulty in verifying t h a t fl is a g a i n a norm.

W e shall write supj~1{~j} for ft. This operation h a s all the usual properties of a su- p r e m u m . I n particular, sup (~s} is a l w a y s defined for finite sets J .

Suppose {aj; ~ E J } is again a n indexed f a m i l y of e l e m e n t s of ~ ( E ) , b u t assume this t i m e t h a t supjej(~*} is defined (in ~(E*)). I n t h a t case, we set

i n / { ~ } = (sup ~7)*.

I n c o n t r a s t to t h e sup operation, it is n o t t r u e in general t h a t i n f ( ~ 9 } ( x ) = inf {~9 (x)}, n o t e v e n w h e n the indexing set J is finite.

W e conclude this section with a description of some relations b e t w e e n n o r m s a n d lattices which will be used l a t e r in s t u d y i n g q u a d r a t i c forms.

L e t ~ be a n o r m a n d let L = = ( x E E ] ~ ( x ) < I } . T h e n L~ is a lattice, a n d is the largest lattice on which t h e values of g do n o t exceed 1. W e denote b y [~] t h e n o r m of t h e lattice L=. I t is clear f r o m t h e definition t h a t ~ ~ [~] ~ q~. [~] m a y also be c h a r a c t e r i z e d as in/{fl E s ~</3}.

P R O P O S I T I O N 1.5.

(a) ~ < fl ~ [~] <

[fl].

(b) [qg] = q[~].

(c) o~(x)=in/q-t[~o~](x), the in/ being taken over all t or over the set O 4 t ~ < l . (d) /or each x E E, the /unction o/ t defined by [qto~] (x) is continuous /rom the left.

THE SPACE OF p-).DIC NORMS 1 4 3

Proo/.

(a) I f a~< fl, t h e n L~ D La f r o m w h i c h we f i n d [a] ~< [fl].

(b) W e h a v e L q ~ = z~L~ a n d h e n c e [qa] = q[a].

(c) W e n o t e first t h a t qto~<<. [qta] so t h a t a ~ < q - t [ q t a ] . H e n c e inf q-t[qta] exists, a n d a<infq-t[qta]. I t follows f r o m (b) a b o v e , t h a t q t[qta] is p e r i o d i c in t w i t h p e r i o d 1 so t h a t inf o v e r all t is t h e s a m e as inf o v e r t h e i n t e r v a l 0 ~< t ~< 1. Now, l e t x b e a n y n o n - z e r o e l e m e n t of E , a n d set a ( x ) = q -t'. T h e n , [qt'a](x)=l, so t h a t q t'[qt~ot](x) = a(x). H e n c e ,

a(x) = inf q-t [qt a] (x), w h i c h p r o v e s (c).

(d) I f x # 0 , t h e n one r e a d i l y verifies t h a t [qta](x) is t h e s m a l l e s t i n t e g r a l p o w e r o f q w h i c h is n o t less t h a n qta(x). C o n t i n u i t y of [qta](x) f r o m t h e l e f t follows i m m e - d i a t e l y .

I t s h o u l d b e r e m a r k e d t h a t as a c o n s e q u e n c e of (a),

[qta]

L]~MMA 1.6. Let a t E s /or 0 ~ < t ~ < l be such that:

(1) at is monotonic increasing in t.

(2) a : = qao.

Then there is a basis o/ E with respect to which all at are canonical.

Proo]. L e t It b e t h e l a t t i c e { x l a t ( x ) ~ < l }. T h e n

(b) 11= g/o,

so t h a t also 1 o ~ l, D 7d o.

N o w lo/Td o is a n n - d i m e n s i o n a l v e c t o r space o v e r ~ / ~ , a n d lt/xd o f o r m a de- s c e n d i n g f a m i l y of s u b s p a c e s of lo/gl o. W e m a y i m b e d t h e f a m i l y {lt/~lo} in a m a x i m a l d e s c e n d i n g c h a i n lo/~lo= V : ~ V z ~ . . . ~ Vn+:=O, w i t h d i m Vt/Vi+:= 1. F o r e a c h t, l~/Td o = V,, for s u i t a b l e i. L e t Yl . . . y~ be a basis for V: chosen so t h a t y,, yi+: . . . . , y~

is a basis for Vt. L e t xl, ..., x~ b e e l e m e n t s of l 0 such t h a t x, m a p s o n t o y, in t h e h o m o m o r p h i s m lo-->lo/Td o = V 1. C l e a r l y x 1 .. . . , x~ is a set of free g e n e r a t o r s of 1 o. F u r -

+ n

t h e r m o r e , for e a c h t, t h e r e is a n i n d e x i such t h a t It = ~lo ~j=t ~ x j , a n d t h e r e f o r e {~x: . . . gx~-i, x~ . . . x~} is a set of free g e n e r a t o r s for lt. I t is n o w clear t h a t all t h e at a r e c a n o n i c a l w i t h r e s p e c t t o t h e b a s i s {x,}.

144 O. G O L D M A N A N D N . I W A H O R I

P R O I ' O S I T I O N 1.7. Let, /or all real t,

actEs

(a) tl < t 2 ~ t . < ~ act.

(b) a t + l = qat

(c) /or each x, act(x ) is continuous ]rom the le/t.

T h e n there ks a unique o~e ~l such that ~ t = [qt~].

Proo/. B y l e m m a 1.6 a b o v e there is a basis (x,} with respect to which all ~t a r e canonical. W e h a v e t h e n ~t ( 5 a,x,) = sup, {q~(t)$a,I}, where

1. /z~(t) e Z

3. /x,(t + 1) =/t~(t) + 1.

N o w define d , = inf {/~, (t) - t} = inf {/z, (t) - t},

0~<t~<l

set r~=q d~ a n d ~ ( ~ a , x , ) = s u p ( r , la,[ }. T h e n a e ~ / , a n d we shall show t h a t [qt~]=act.

I t is clearly sufficient t o consider o n l y 0 ~< t ~< 1. W e note also t h a t from condition (c), d~ = bt~ (t) - t for some t with 0 ~< t < 1.

We h a v e [qtac]=inf{fles F r o m t h e definition of zr we have ~<~q tact, so t h a t [qt~] < ~tt. Hence we m u s t show t h a t act<.. [qt~].

N o w [qtac] is the n o r m of t h e lattice

{xl~(x)<<.q-').

Suppose t h e n t h a t ~ t ( x ) > 1. Since act EE, t h e values assumed b y act on t h e non- zero elements of E are integral powers of q. Hence we have ~Zt(X)>~ q. ^{W i t h} x = ~ a i x t , there is an index i such t h a t q~'~t~la~l>~q. Now, for some t' with 0 ~ < t ' < l , we h a v e d~ = / x , ( t ' ) - t ' . W e consider

q~l.,l=r162 [a,[.

Suppose first t h a t t ~< t'. Then, /~i(t) ~</x, (t') so t h a t q~, [a,l 1> qm,t,-t" la, I >~ q~-t" > q-t.

On t h e other hand, suppose t h a t t ' < t . Then,

/x~ (t) <~ tz, (1) = ~ (0) + 1 < tt~ (t') + 1, a n d hence qn, lag I ~> q,,<t,-x t.[a ' I >~ q-t" > q-t.

Thus, in either e v e n t ac(x) >1 qd'la,] > q-t.

This shows t h a t act <~ [qt~] a n d hence act = [qtac]. The uniqueness of ac follows f r o m P r o p o - sition 1.5.

THE SPACE OF O-ADIC NORMS 1 4 5

Remark. I t follows f r o m the a b o v e t h a t t h e r e are a t m o s t n distinct n o r m s a m o n g the [qta] for 0 ~ < t < l .

L e t E = E I + E 2 (direct sum), a n d let ~ET/. W e shall s a y t h a t E 1 a n d E 2 a r e orthogonal with respect to a if a ( x l + x2)= sup (:r , ~(x2)), w h e n e v e r xt e e l .

LEMMA 1.8. Let E = E I + E 2 (direct sum) and let ] be the projection o / E onto E 1.

Let ~ e 7~. Then E 1 and E 2 a r e orthogonal with respect to ~ i], and only i/, ~(x)>~ :r /or all x e E.

Proo/. Suppose E 1 a n d E 2 are o r t h o g o n a l with respect to :r T h e n ~ ( x ) = sup (~(/(x)), a ( x - / ( x ) ) ) , hence ~(x) ~ ~(/(x)). Suppose n o w t h a t a(x) ~> o:(/(x)) for all x e E . Since ~(x) < sup (~(/(x)), ~ ( x - / ( x ) ) ) only if a(/(x)) = o~(x-/(x)), it follows t h a t a(x) = sup (~(/(x)), ~ ' . x - / ( x ) ) ) , so t h a t E 1 a n d E 2 are o r t h o g o n a l w i t h respect to g.

COROLLARY 1.9. Let E = E I + E 2 (direct sum) and let a E ~ . Then E 1 and E 2 are orthogonal with respect to ~ if, and only i/, E 1 and E~ are orthogonal with respect to [qt ~] /or all t.

Proo/. L e t / be t h e projection of E o n t o E 1. T h e n a(x)>~ ~ / ( x ) ~ [ q t ~ ] ( x ) ~ [qt~] (/(x)) for all t a n d the result follows f r o m L e m m a 1.8.

Section 2. T h e topological structure o f T / ( E )

W e introduce a metric in t h e set 7~ of all n o r m s on E. I f ~, fl E T/ consider, f o r non-zero x, t h e q u o t i e n t o~(x)/fl(x). As we h a v e a l r e a d y seen, this defines a contin- uous function on the c o m p a c t space P(E). As ~ ( x ) > 0 for non-zero x, t h e f u n c t i o n [log (~(x)/fl(x))[ is b o u n d e d on P(E), a n d we define

d(~,, fl) = x.0sup log fl~)l'~(x) T h e r e is no difficulty in verifying t h a t d is a metric on ~ .

There is a n explicit f o r m u l a for d which will be useful later.

P R O P O S I T I O N 2.1. Let ~ and fl be norms such that ~ is canonical with respect to a basis {x~} and fl is canonical with respect to {y,}, Then,

[lo fl(xi) .

a(y,)]

Proo/, I t is clear f r o m t h e definition of d t h a t

d(~, fl) = s u p / s u p 1o- fl(x) , =(Y)I

146 O. G O L D M A N A N D N. I W A H O R I

Now, fl(x)_ , fl(~ aixi)

sup log - ~ - sup log a - ~ ~ '

w i t h t h e s u p r e m u m being t a k e n over all a 1 .. . . . an E K, n o t all 0. Hence, sup/7(x) >>- sup/7(x,)

L e t m = o~(~a,x,) = sup,

(la, I

I ,1

Hence,

/7(~ a,x,)<~ sup _,

(la, I/7(~,)) < sup (m

so t h a t

/7(~ aixt) /7(x~)

~(~ aixi~) <~

T h u s sup log/7(x) fl(xi)

xEE - - ~ = sup, log ~(Xi)"

I n the same way, we o b t a i n sup log ~(y)

yEE ~(~

a n d t h e assertion follows immediately.

~(Yr

= sup log - - ,

s /7(Yr

F r o m n o w on, when we speak of t h e t o p o l o g y of 71 it will be u n d e r s t o o d t o m e a n the metric t o p o l o g y defined b y d.

THEOREM 2.2. 71 is complete.

Proo/. L e t {an} be a C a u c h y sequence in 71. L e t /7 be a n y element of 71, a n d set /,~(x) = l o g (~n(x)//7(x)), for x # 0 in E. T h e n the /n are continuous functions on P(E) a n d form a C a u c h y sequence in the uniform t o p o l o g y on P(E). Hence g(x)= limn_>~/n (x) exists, where g is continuous on P(E), a n d t h e convergence is uniform.

Set ~(x)=/7(x) e ~(x) on E, defining ~ ( 0 ) = O. Then, for x # 0 we have ~ ( x ) > O, also o~(ax)=fala(x) for a E K . We have log (Ct(X)/~n(X))=g(x)--/n(X), SO t h a t

Iim I log ~(x) - log a , (x)[ = O, u n i f o r m l y for non-zero x.

N o w suppose t h a t for some x a n d y we h a v e a ( x + y ) > s u p (a(x), ~(y)). Clearly t h e n none of t h e elements x, y, x + y is O. Choose e > 0 such t h a t E < 89 (log ~ ( x + y ) - log ~(x)) a n d also s < 89 (tog zc(x + y) - log a(y)). Then, there is an n such t h a t log co(z) - log ~n(z)l < e for all non-zero z. I n particular,

THE SPACE OF ~0-ADIC NORMS 147

log a(x + y) - log an (x + y) <

- log a(x) + log O~n(X) <

a n d therefore, log a~ (x + y) - log an (x) > 0, or an (x + y) > a~ (x). I n e x a c t l y the s a m e way, we find ~ n ( x + y)>~n(Y). B u t this contradicts t h e fact t h a t a~ is a n o r m . Thus, we m u s t h a v e ~(x+y)<~sup(~(x), ~(y)), a n d hence a is a n o r m . I t follows i m m e d i a t e l y f r o m t h e definition of ~, t h a t limn~ccd(~, a ~ ) = 0 . This completes t h e proof t h a t T / i s complete.

T ~ E O R E M 2.3. T/ is locally compact. More precisely, i/ ~ e T l , and b > 0 , then

{fle ~tld(~, fl) < b} is ~o~pact.

Proo/. L e t D = {fl fi ?l[d(~, fl) < b). B y T h e o r e m 2.2, D is complete. L e t X be a subset of E h a v i n g t h e following properties: 0 ~ X , X c o m p a c t , a n d if x is a non-zero e l e m e n t of E, t h e n a x e X for some a e K . Such a set X m a y be o b t a i n e d in t h e following m a n n e r : let (xi) be a basis of E, a n d set X = { 5 aix,] sup {[a,I} = 1}.

D e n o t e b y F t h e space of functions on X described b y the restriction to X of log fl, w i t h f l e D. T h e metric t o p o l o g y in D coincides w i t h t h e u n i f o r m t o p o l o g y in F . W e shall show t h a t F consists of u n i f o r m l y bounded, u n i f o r m l y equicontinuous func- tions on X.

Because, for f l e D , d(~, f l ) < b, we h a v e

e-b~(x) < fl(x) <. e%(x), for all x e X , so t h a t

l log fl(x) I < b + l log ~(z) l 9

Since log a(x) is b o u n d e d on X, it follows t h a t t h e elements of _P are u n i f o r m l y b o u n d e d on X .

Also, since zc(x) is b o u n d e d a w a y f r o m 0 on X, all fl e D are u n i f o r m l y b o u n d e d a w a y f r o m 0, so t h a t the uniform e q u i c o n t i n n i t y of (log fl} is t h e s a m e as u n i f o r m e q u i c o n t i n u i t y of (fl}. Now,

I f t ( x ) - fl(Y) I ~< fl(x - y) <~ e%t(x - y),

a n d because ~ is u n i f o r m l y continuous on X, t h e result follows. H e n c e b y t h e theo- r e m of Ascoli-Arzela, F is c o m p a c t , a n d hence D is also c o m p a c t .

W e con~ider n o w the connectedness properties of ~/.

T H E O R ~ M 2.4. Let ~, fl be two norms, and let 0~<t,.<l. Let P(t) = (9r e ~ l ~(x) <~ a(x)l-~ fl(x) t, all x e E}.

148 O. GOLDMAN" XND N. IWAHORI

Then, P(t) is not empty. Set gt = sup { 7 [ ? tiP(t)}. Then, :~t has the [ollowing properties:

(a)

:~l=fl

(b) d(Tet., :zt,) = It1 - t2l d(~, fl)

(c) I[ {x,} is a basis with respect to which both o~ and fl are canonical, then ~t is also canonical with respect to {xi}, and ~t(xi)=o~(xi)l-t fl(x~) t.

Proo/. Let {xi} be a basis with respect to which b o t h ~ a n d fl are canonical.

Define yt by:

yt ( 5 a, x,) = sup {] a, ] a(x,) 1-t fl(z,)t},

i

so t h a t ytE ~ . W e shall show first t h a t y t E P ( t ) . W e h a v e la,] a(x,) <~ o~(~ a,x,), so t h a t

la, II-t at(x,) 1-t ~< a(% a,x,) 1-t.

Similarly, I a, [tfl(x,) t 4 fl(5 a, x,) t ,

s o t h a t

]a,

Since this is so for all i, it follows t h a t ytEP(t). F u r t h e r m o r e , for each x, y(x) is b o u n d e d f r o m above, as y ranges t h r o u g h P(t), so t h a t ~t is defined. W e have, of course, Yt ~ ~t.

W e shall n o w show t h a t a c t u a l l y yt =ret. N a m e l y ,

~t (x~) < a(a)~-t fl(x~) t,

so t h a t ~ t ( S a , xi) <~ sup ([a, [ ~t (xi)) 4 sup ([a,] ~z(x,)~-tfl(x~) t) = Y t ( 5 a,x~).

Thus, zeta< yt or ztt = y t - This p r o v e s assertion (c).

T h e assertion (a) is obvious, while (b) follows i m m e d i a t e l y f r o m t h e f o r m u l a of Proposition 2.1.

As a n i m m e d i a t e corollary, we have:

COROLLARY 2.5. T/ is arcwise connected.

Proo[. W i t h :r a n d fl given, c o n s t r u c t ~t as above. I t follows f r o m (b) t h a t t h e m a p t-->zt is a continuous m a p p i n g of t h e unit i n t e r v a l into ~/. Since n o = a a n d re 1 =fl, the result follows.

P R O P O S I T I O ~ 2.6. T/ /8 contractible.

Proo/. L e t ~r E 7// be given. I f a E ~ , let ret(~) be t h e arc described b y T h e o r e m 2.4, with r~0(~)---~ o a n d g~(a)=ar L e t 1 be t h e closed i n t e r v a l 0~<t~<l, a n d define

THE SPACE OF p-ADIC ~ORMS 149 / :/• b y /(t, a) = ~ t ( ~ ) - Then, riO, ~) = a 0 while /(1, a) = ~ . Thus, t h e result will follow w h e n we p r o v e t h a t / is continuous.

W e h a v e d(/(tl, ~1),/(t~, r162 =d(~t,(~l), gt2(a2)

so t h a t d(/(tl, ~i),/(tu, ~2)) < d(a~t, (~1), 7Q, (0c2)) + ]t I - t21 d(~2, a0).

W e shall n o w show t h a t

d(Tet(a), xet(fl)) <~ td(~, fl), f r o m which e v e r y t h i n g will follow.

L e t {x~} be a basis w i t h respect to which b o t h :% a n d ~ are canonical, {yj} a basis with respect to which a• a n d fl are canonicall Then, gt(a) is canonical w i t h respect t o {x~}, while ~t(fl) is canonical w i t h respect t o {yj}. F u r t h e r m o r e ,

~t (~) ( x 0 = ~o (x~) 1-t ~(x~)~

~ ( ~ ) (yj) <<. ao(yj)~-~a(yj) t

~ ( ~ ) (x,) < ~o (x,) ~-~ fl(x,) ~ a~t (fl) (yj) = O~o (yj)l-t fl(ys) t.

Combining these relations with t h e f o r m u l a of P r o p o s i t i o n 2.1 gives i m m e d i a t e l y d(xet(a), xet(fi)) <. td(~, fl). This completes t h e proof.

L e t ~ e ~ . D e n o t e b y C(~) t h e set of real n u m b e r s {log a(x) l x ~ E , x~=O}. Choose a basis {x~} w i t h respect to which :r is canonical. T h e n

ot(~aiz,) ⁼ sup (m, Jail),

a n d for each non-zero x ~ E , we h a v e log a ( x ) = l o g m ~ + l o g [a], for some a ~ K * , some i.

T h i s shows t h a t C(a) is the union of a finite n u m b e r of cosets of R / Z log q, the n u m b e r of cosets being no m o r e t h a n n = d i m E. W e shall call t h e n u m b e r of these cosets t h e rank of a, a n d shall denote it b y r(a). I n t h e n e x t section we shall associate multi- plicities to the cosets of C(~).

T h e set C(a) is u n i f o r m l y discrete in t h e following sense: there exist positive real n u m b e r s d such t h a t , if r, s e C(a) a n d [ r - s I ~< d, t h e n r = s . W e shall call such n u m b e r s d separating numbers of ~.

P R O P O S I T I O ~ 2.7. Let a ~ ' l , and let d be a separating number of a. I/ f l ~ l and d(o~, fi) <~ 89 then r(fl) >~ r(a).

159 O. G O L D M A N A N D N . I W A H O R I

Proo/. Let {x,} be a basis with respect to which b o t h ~ a n d /3 are canonical, cr a , x , ) = s u p (m, la,]), fl(Xa, x,) = s u p (m; la, I).

Then, P r o p o s i t i o n 2.1 shows t h a t d(~,/3) = sup [log m~ - log m~ I . Hence, [log m, - log m; I ~<

89 l<~i<~n.

L e t r ( ~ ) = r, a n d r e n u m b e r the x~ so t h a t log m I . . . log mr are i n c o n g r u e n t m o d Z l o g q . F r o m the choice of d as a s e p a r a t i n g n u m b e r of ~, it follows t h a t log m~ . . . log mr are i n c o n g r u e n t rood Z log q, so t h a t r(/3) ~> r = r(a).

COROLLARY 2.8. The set o/ / 3 E ~ /or which r(/3)=n is open and everywhere dense in ~.

Proo/. Proposition 2.7 shows i m m e d i a t e l y t h a t t h e set u n d e r consideration is open.

T o see t h a t t h e set is e v e r y w h e r e dense, let ~ E ~ a n d let {x~} be a basis w i t h re- spect to which ~ is canonical. Set mi =a(x~). L e t s > 0 , a n d choose real n u m b e r s t~ as follows:

[ t t I < e , a n d t l + l o g m 1 . . . t n + l o g m n are incongruent m o d Z log q. N o w p u t

/3(~ a,x,) = sup

(et'm,

Then, /3 E ~ a n d d(a,/3) = s u p ]t~] < e. Finally, it is clear t h a t r(/3)=n. Thus, t h e set of n o r m s of r a n k n is e v e r y w h e r e dense in ~ .

TH]~OREM 2.9. Let a E ~ have rank n, let d be a separating number o/ o~, and let {x~} be a basis with respect to which a is canonical. Then, i/ d(a,/3) <~ 89 /3 is also canonical with respect to {xi}.

Proo/. Set a(x,) = m , . Since r(a) = n , a n d a(Za~x~) = s u p (m,[a~I), t h e n u m b e r s log m 1 . . . log mn are i n c o n g r u e n t rood Z log q. Because d(a,/3) ~< }d, we h a v e [log/3(xi) - log m~I~< 89 a n d since d is a s e p a r a t i n g n u m b e r of a, it follows t h a t log/3(xl) . . . log/3(x,) are incongruent m o d Z l o g q . L e t x = ~ a i x i be a n y non-zero e l e m e n t of E . F r o m t h e incongruence, m o d Z l o g q , of {log/3(x~)}, we find t h a t those of the n u m b e r s /3(x~)]a,I which are non-zero are distinct. H e n c e

fl(X a,x,) = sup (fl(x,) [at I), or /3 is canonical with respect t o {x~}.

COROLLARY 2.10. Let ~ ' be the set o/ all o~E~ with r ( ~ ) = n . Then ~ ' is locally Euclidean o/ dimension n.

T H E SPACE OF ~0-ADIC NORMS 151 Proo/. L e t ~ E Tl', let d be a s e p a r a t i n g n u m b e r of :r a n d let {x~} be a basis w i t h respect t o which ~ is canonical. Set D = {flld(~, fl)< 89 T h e n n c T/' is a neighbor- hood of ~, a n d T h e o r e m 2.9 shows t h a t each element of D is canonical with respect.

to {x~}. Define / : D - - > R n b y

l(fi) = {log fl(xa) . . . log fl(xn)}.

T h e n ] is a h o m e o m o r p h i s m of D with t h e open s u b s e t of R ~ defined b y {(r 1 .. . . . r n ) ] i r ~ - l o g ~(x~)] < 89

P R O P O S I T I O N 2.11. Let {xi) be a basis o/ E . Let A be the set o / a l l norms which are canonical with respect to (x~). Then A is a closed subspace o/ 74.

Proo[. Suppose ~ ~ A. T h e n there are a~ E K, n o t all 0, such t h a t sup {log ~(a,x~) - log :r a~x~)} = d > O.

N o w suppose t h a t f i E ~ such t h a t d(~,fl)<~ 89 Then, for all non-zero x, we h a v e 1 d a n d combining this with the definition of d gives

]log ~(x)

sup {log fl(a,x,) - log fi(~ a,x,)} <~ ~d.

T h u s also flr A which shows t h a t A is closed.

COROLLARY 2.12. Let ~ ' be again the set o/ a E ~ with r ( a ) = n , and let X be a connectedness component o/ ~ ' . Let {xi} be a basis with respect to which some element o/ X is canonical. Then every element o/ X is canonical with respect to {x~}.

Proo[. L e t A be the set of all n o r m s which are canonical with respect to {x~}.

T h e n b y Proposition 2.11, 7//'0 A is a closed subset of ~ ' . H o w e v e r , T h e o r e m 2.9 shows t h a t ~ ' N A is open in ~ ' . T h e assertion follows i m m e d i a t e l y .

Section 3. The action of Aut ( E ) on 7~(E)

W e denote b y A u t (E) t h e group of linear a u t o m o r p h i s m s of E w i t h its n a t u r a l locally c o m p a c t topology. A u t (E) acts on ~ ( E ) as follows: if a E A u t (E) a n d ~ E ~ ( E ) , t h e n a ~ ( x ) = ~ ( ( r - l x ) . I t is clear f r o m t h e definition of t h e metric on 71, t h a t t h e e l e m e n t s of A u t ( E ) are r e p r e s e n t e d b y isometries of 7//. I t is also clear t h a t the subset.

g of ~ is stable u n d e r the action of A u t (E).

152 O . G O L D M A N A N D N . I W A H O R I

P R O r O S I T I O ~ 3.1. The kernel o/ the representation o/ A u t ( E ) by trans/ormation~, o/ ~(E) consists o/ the automorphisms x--> cx, with

I cI

Proo/. If for some x E E the elements x and a(x) are linearly independent, then there exist norms ~ so t h a t a(x)~: ~(ax). Hence x and a(x) are linearly dependent fol all x E E , from which it follows t h a t a(x)=cx with c a fixed element of K. As we have a(cx)=[cia(x), we conclude t h a t ]c] = 1 . Conversely, it is obvious t h a t a n y au- tomorphism of the form x--> cx, [c] = 1 is represented b y the identity on ~ .

I f ~ E ~ , we denote b y G~ the isotropy group of ~ in Aut (E).

P R O r O S I T I O ~ 3.2. G , = ntGcqt, 1.

Proo/. The assertion is an immediate consequence of the definition of [a], and Proposition 1.5 (c).

COROLLARY 3.3. For each :r ~(E), the isotropy group G~ is a compact open sub- group o/ Aut (E).

Proo/. Let L be a lattice and fl its norm. I f {x,} is a set of free generators of L and a is in A u t ( E ) with axi=~a,jxj, then aEG~ if, and only if, all atj are in and det (a,j) is a unit in ~ . I t follows immediately t h a t G~ is a compact open sub- group of Aut (E) (and is in fact a m a x i m a l compact subgroup).

If a is a n y norm, then b y the renhark following Prop. 1.7, only finitely m a n y of the norms [ q t ] are distinct. Hence, because of Prop. 3.2, G~ is also a compact open subgroup of A u t ( E ) .

Remark. Proposition 2.1 m a y be used to give an explicit description of the ele- m e n t s of G~. L e t {x,} be a basis with respect to which a is canonical. Let a E Aut (E), a n d suppose a ( x i ) = ~ j a z x j. Then, aEG~ if, and only if,

[at, l<cc(x~)o~(xj) -1, i , i = 1 , 2 . . . n a n d I det (aj~)] = 1.

COROLLARY 3.4. The map A u t ( E ) • ~(E) is continuous.

Proo/. We h a v e

d(afl, r~) ~< d(afl, aoO + d(aa, r~) = d(fl, o:) + d(o:, a-iron).

Hence, ff a - l r E G~, we obtain d(afl, ra) ~ d(fl, a). Since G, is open, the assertion follows.

T H E S P A C E O F p - A D I C N O R M S 1 5 3

P R O P O S I T I O N 3.5. Let :r and fl be elements o/ )l, and let d be a positive real number. Let X~ be the orbit o / ~ under the action o / A u t (E), and let D = {r e ~[d(~, r) ~ d}.

Then X~ N D is finite.

Proo/. L e t {xi} be a basis with respect to which b o t h :r a n d fl are canonical.

,Set ~(xi) = r, a n d fl(xi) = el. Suppose a e A u t (E) is such t h a t a~ E D. Set a-l(xi} = ~. a,j xj. T h e n ao~(x,) = sup {rjla,, ]}.

S 1

F r o m t h e fact t h a t d(a~,fl)~< d, we get

log fl-(~-) I ~< d

a n d therefore e- as, < sups { rs lai s]} ~< e a s,.

H e n c e [ais]<<.edsi/rj. I t follows t h a t t h e set of a E A u t ( E ) for which a : c E D lies in a c o m p a c t subset of M~ (K). However, t h e m a p A u t ( E ) - - > }i defined b y a--> a~ is con- tinuous. Hence the set {a e A u t (E) I a~ e D} is a closed subset of A u t (E), so t h a t it is compact. Since G~ is an open subgroup of A u t (E), it follows t h a t X~ N D is finite.

As an immediate consequence, we have:

COROLLARY 3.6. The orbit X~ is a closed discrete subset o/ ~ .

L e t ~ E ~ a n d ~ {x,} be a basis with respect to which :r is canonical. Set

~(xi) = m,. Then, t h e n u m b e r s log m 1 .. . . . log mn are representatives (with possible repe- titions) of t h e cosets of C(~) m o d Z l o g q . L e t r=r(ac), a n d suppose t h e x, h a v e been r e n u m b e r e d so t h a t log m 1 .. . . , mr are i n c o n g r u e n t m o d Z log q. Let, for 1 ~< i ~ r, E, be t h e space spanned b y all xj which are such t h a t log m j - - l o g mi (mod Z log q). Then, E is the direct sum of E 1 .. . . . Er. F u r t h e r m o r e , there is a lattice n o r m 2, E s (E,) such

t h a t t h e restriction of ~ to E, is m, 2,.

The dimension of the space El does n o t d e p e n d on t h e choice of t h e basis {xi}.

N a m e l y , suppose E = E ~ + ... +E'~ (direct sum). Suppose f u r t h e r t h a t there is a

9 / / p I 9

2 s e s a n d a positive real n u m b e r mj such t h a t ~ l E ~ = m i 2 , a n d log mt ~ log m, ( m o d Z l o g q ) . L e t ]i be t h e projection of E onto E,, a n d let x be a non-zero ele- m e n t of E~. W e h a v e ~(x)=m~2~(x), so t h a t log~(x)~--logmi(modZlogq). At t h e same time, t h e ~ non-zero n u m b e r s a m o n g mj2j] s (x) are distinct, hence ~(x) = supj (mj2j]s(x)), so t h a t in particular, [,(x)=~0. Thus, t h e restriction of /, t o El m a p s ^P" E" , m o n o m o r - phically into E,, so t h a t dim E[ ~< dim E~. F r o m t h e s y m m e t r y , it follows t h a t dim E~ =

] 1 - 6 3 2 9 1 8 . A c t a mathematica 109. I m p r i m 6 le 13 j u i n 1963.

154 O. G O L D M A N A N D N . I W A t t O R I

dimE~. W e shall call dimE~ t h e multiplicity of t h e coset (in C(a)) which is repre- sented b y logm~. We shall also use t h e n o t a t i o n C~(~) to denote t h e set C(~) t o g e t h e r with the assigned multiplicities. I t is clear t h a t t h e sum of t h e multiplicities of the:

cosets which comprise C(~) is n.

THEOREM 3.7. Let o~ and ~ be elements o/ 71. Then, a necessary and su//icient condition that r162 and ~ be conjugate under the action o/ A u t ( E ) is that C,, (~)= Cm (~).

Proo/. Suppose first t h a t fl = a~, with a E A u t (E). L e t (x~} be a basis with re- spect t o which ~ is canonical, a n d let y~ =a-lx~. T h e n fl is canonical with r e s p e c t t o (y~}, a n d ;6(y,)=~(x,). I t is clear t h a t Cm ( a ) = Cm(~).

N o w suppose Cm(~)= Cm(fl). Let (x~} be a basis with respect to which b o t h a n d fl are canonical, a n d f u r t h e r m o r e such t h a t

1 ~< a(Xl) < a(x2) < . . . a(xn) < q.

T h e n each ~(x~) occurs in t h e set (:~(x 1 . . . :r as often as t h e multiplicity of t h e associated coset in C(a). T h e n u m b e r s fl(x~) are n o t necessarily in increasing order~

However, there is an element a E A u t (E), such t h a t with ~, = a~, we have 1 < r(xl) < r ( x , ) <<... r(xn) < q.

F u r t h e r m o r e , a is a p r o d u c t of a diagonal t r a n s f o r m a t i o n with a p e r m u t a t i o n of t h e x~, so t h a t ~, is still canonical with respect to (xi}. As we h a v e Cm (~')= Cm (fl)--C,~(a), it follows t h a t ~(x~) = ~(x~) all i. Hence ~ = a, or ~ a n d fl are c o n j u g a t e u n d e r A u t (E).

COROLLARY 3.8. The orbit space A u t ( E ) \ ) ~ ( E ) is naturally homeomorphic to the symmetric product o/ n circles.

Proo/. Set T = R / Z l o g q so t h a t T is a circle, a n d let S be the s y m m e t r i c p r o - d u c t of n copies of T. We define a m a p h:~(E)-->S as follows: given ~ E ~ ( E ) , C(:r is a subset of T consisting of r(g) points, each with an assigned m u l t i - plicity, the s u m of the multiplicities being n. I f t 1 . . . tr are those points of T, a n d vl . . . . ,vr are their multiplicities, set

h(o~) = (tl, t 1 .. . . . tr), each tt appearing ~ times.

I t is clear t h a t h m a p s 7/(E) o n t o S. I t follows f r o m T h e o r e m 3.7 t h a t t h e inverse.

image of a point of S is precisely one orbit of 7/(E) u n d e r t h e action of A u t (E).

Hence h defines a 1 - 1 m a p h' of A u t ( E ) \ 7 / ( E ) o n t o S. To show t h a t h' is a h o m e o - morphism, we shall show t h a t h is a continuous open map.

THE SPACE OF ~-ADIC NORMS 155 W e show first t h a t h is a n open m a p . L e t ~ 6 ~ / , let d > 0 be given, a n d set D = {fl I d ( ~ , ~ ) < d}. L e t {x,} be a basis w i t h respect t o which ~ is canonical a n d such t h a t

1 ~< ~(xl) ~< ~(x2) ~<... < ~(x~) < q.

Suppose t h a t (t 1 . . . tn) E S is such t h a t tl ~< t 2 ~<... ~< t~, I t s - log ~(x~)l < d. Define f l e T/

b y

~(~.

x,)

T h e n d(~, fl) = sups {] ts - log ~(xs)[} < d,

so t h a t flED, while h(fl)=(t 1 ... tn). T h u s t h e i m a g e of D contains a n open set, which contains h(a), so t h a t h is a n open m a p .

T o see t h a t h is continuous, let a, f l q ~ w i t h d(a, fl)=d. L e t {xs} be a basis with respect t o which b o t h a a n d fl are canonical. T h e n

[log a(xt) - log fl(xd] ~< d.

Hence, b y choosing d sufficiently small, we can m a k e h(fl)lie in a n y prescribed n e i g h b o r h o o d of h(a). Thus, h is continuous a n d t h e p r o o f is complete.

Remark. T h e s t r u c t u r e of t h e orbit space A u t ( E ) \ ~ ( E ) m a k e s it possible t o de- fine a n i n t e g r a t i o n on T/(E) which is i n v a r i a n t u n d e r t h e action of A u t ( E ) . N a m e l y , let [ be a c o m p l e x - v a l u e d f u n c t i o n on ~/(E) which is continuous with c o m p a c t sup- port. L e t a E T/, a n d define for a E A u t (E), g(a, a) = [ ( ~ ) . Then, g is a continuous function on A u t ( E ) ( ~ is fixed) w i t h c o m p a c t s u p p o r t . T h e c o n t i n u i t y of g is i m m e - diate. Suppose D is t h e s u p p o r t of [. Then, since t h e o r b i t of a is discrete, t h e r e are only a finite n u m b e r of n o r m s ~a in D. Hence, g vanishes outside a finite union of eosets m o d G~, a n d since G~ is c o m p a c t , it follows t h a t g has c o m p a c t s u p p o r t . W i t h a H a a r m e a s u r e chosen on A u t (E), f o r m

h(a) = fAut(s) g(a, a) da.

Then, h depends o n l y on t h e o r b i t of ~ u n d e r A u t (E), so t h a t it defines a function on t h e s y m m e t r i c p r o d u c t of n circles. T h e o r d i n a r y p r o d u c t Tn of n-circles, being a group, carries a H a a r measure. W e lift ~ to a function h' on Tn, a n d t h e n define STn h' t o be t h e integral of ] on ~/.

W e h a v e shown a b o v e t h a t Cm (~) gives a complete set of i n v a r i a n t s for equi- valence of n o r m s with respect to the g r o u p A u t (E). We now consider the question

156 0 . G O L D M A N A N D N . I W A H O R I

of equivalence with respect to the elements of the isotropy group G~ of a preassigned norm ~. I n order to describe the invariants for this type of equivalence we m u s t first consider the extensions of norms from E to the exterior algebra of E.

We denote b y A E the exterior algebra of E, b y A r E the homogeneous compo- nent of A E of degree r. I n particular, A 0 E = K and A l E - - E . I f a E T/(E), we shall associate to c r norm A r a on A r E , for r~> 1.

L e t 2 be an element of the dual space E* of E. Then there is defined a deri- vation d~ of A E which m a p s A r E into A r - I E , and which coincides with 2 on E.

Hence, if (oEA~E and ~1 . . . ~r are elements of E*, then d ~ , . . . d ~ ( a ) ) E K . For E ~ (E), we define, for co EAr E,

IDA....

as ~1 . . . ~tr range independently over the non-zero elements of E*. As we have al- r e a d y seen in section 1, the existence of such a supremum lies in the compactness of the space P(E*). I t follows from Prop. 1.2 t h a t A x a = a .

I t follows without difficulty from the definition that, for r~>l, and w EAr+~ (E), we have

A~ (~) (d~ ~o) A,+I (~) (~o) = sup

~o~* ~*(~)

We leave to the reader the details of the verification of the following properties of Ar (a).

P~OI"OSlTION 3.9. I / ~ is a norm o/ E, then At(a) is a norm o/ A r E . I / a is canonical with respect to the basis {x,} o/ E, then At(a) is canonical with respect to the basis {x~, A ... A x~r}, (i I < i 2 < . . . < it) O/ Ar E, and

Ar (a) (x~, A ... A x~r ) = ~(x~,)...a(%).

I f a EAut (E) and a E ]I(E), then it follows directly from the definition t h a t A r ( a a ) = a r A r ( a ) , where err is the automorphism of A r E induced b y a.

L e t ~ and fl be two norms of E. We define, for l~<r<<.n, the r-discriminant A r(fl,a) of fl with respect to a b y

Ar (fl)(o~)

Ar(fl, ~) = sup

~A~E A~ (a) (o~)

where co ranges over the non-zero elements of A r E . As usual, the s u p r e m u m exists because of the compactness of the projective space P ( A r E ) .

THE SPACE OF p-ADIC NORMS 157 P ~ O P O S l T I O N 3 . 1 0 . Let {xi} be a basis with respect to which both :r and fl are canonical, and suppose that the elements o/ the basis are arranged so that

~(x~) ~ "'" ~(z=)"

Then A~ (/~, ~) = 1~ ~'-'z~".

Proo/. Since A~(a) is canonical w i t h respect t o t h e basis (x~, A ... A % } (i~ < i~ < ... < i~) of A ~ E ,

A~ (~) (x~, A ... A x,,) we h a v e A~ (fl, ~) = sup Ar (~) (x h A ... A xir )'

t h e s u p r e m u m being t a k e n o v e r t h e set of r-tuples (i1< ... < i~). H o w e v e r , we h a v e .. A x, )fl

h~ (~) (x~, A .../~ %) J~l ~(xi~)

I t follows i m m e d i a t e l y f r o m t h e fact t h a t

~(xl)>~(x,) >fl(xn)

~(~) ~)>~"" ~(~)'

t h a t the s u p r e m u m occurs for t h e r-tuple (1, 2 . . . r), a n d hence t h a t

• (t~, ~ = n ~(x~)

Using t h e s a m e n o t a t i o n as above, it follows f r o m t h e p r o p o s i t i o n t h a t t h e n u m - bers fl(xr are d e t e r m i n e d b y t h e discriminants Ar(fl,~ ) a n d do n o t d e p e n d on t h e choice of t h e basis {x~}. Thus, if a a n d fl are t w o norms, a n d {x,} is a basis w i t h r e s p e c t t o which b o t h a a n d fl are canonical, t h e n t h e u n o r d e r e d n - t u p l e {fl(Xl)/a(x 1) . . . fl(xn)/a(xn)} is c o m p l e t e l y d e t e r m i n e d , including multiplicities, b y ~ a n d fl, a n d does n o t d e p e n d on the choice of t h e basis. W e shall call t h a t (unordered) n - t u p l e of real n u m b e r s the invariants of fl with respect to a, a n d shall denote it b y I(fl, a). I f 0~<t~<l we consider also t h e i n v a r i a n t s l(fl,[qta]). Considered as a function of t, the i n v a r i a n t s I(fl, [ q t ] ) will be called t h e elementary divisors of fl w i t h respect to a.

158 O. G O L D M A N A N D N . I W A H O R I

I f a EAut (E), then the definition of I(fl, o~) shows immediately t h a t I(afl, a~)=

I(fl, ~). In particular, the elementary divisors of aft with regard to a~ are the same as those of fl with respect to g.

We shall use the elementary divisors to give a criterion for the equivalence of two norms with respect to t h e isotropy group of a third norm.

L~.~MA 3.10. Let o~ and fl be two norms and let {x~) be a basis with respect to which o~ is canonical. Then, there is a (l E G~ such that aft is canonical with respect to {x,~.

Proo/. L e t {y,) be a basis with respect to which both a and fl are canonical.

According to our description of Cz(a), the set of n numbers {log a(xl) . . . log a(xn)) represent the same cosets, m o d Z l o g q , as do the numbers {loga(yl) . . . log~(yn)}, each coset being represented the same n u m b e r of times in the first set as in the se- cond. Hence, b y renumbering the Yi, we h a v e ~(xi)=qh~a(y~), with h~ fiZ.

Define a E A u t ( E ) b y a(y~)=gh~x~. Then, because a is canonical with respect to (y~), a~ is canonical with respect to {x~). At the same time,

O'a(Xi) = a ( o ' - l x t ) = q ht a ( y t ) ~-~ a ( X t ) ,

and hence a EG~. FinaLly, because fl is canonical with respect to (y~}, it follows t h a t aft is canonical with respect to {x~).

We choose a norm ~ which will be fixed for the rest of this section. L e t {x~}

be a basis with respect to which ~ is canonical, and such t h a t - < a ( x l ) < a ( x , ) < . . . < a(zn) <<- 1. 1 q

This basis will also be k e p t fixed. With respect to the basis {x~), we have [zt] (x~)= 1.

Using the basis {xt) we identify A u t ( E ) with GL(n, K) as follows: to a EAut (E) we associate the m a t r i x (aj~) given b y ax~=~aj~xj.

L e t g be the r a n k of ~, and let vl . . . v a be the multiplicities of the ~(x~).

Namely,

0~(gl) = . . . = 0~(X,,) ( 0~(Xv,+l ) = . . . = g(Xv,+v2) < etc.

We place along the main diagonal of a generic n • m a t r i x square blocks of size v 1, v 2 .. . . . vg. Then, the elements of G~ are the matrices with coefficients in ~ , whose determinant is a unit in ~ a n d whose elements to the left of the aforementioned diagonal blocks are in p. I n passing, we observe t h a t the elements a E A u t ( E ) h a v i n g

T H E S P A C E O F ~ - A D I C N O R M S 159 t h e p r o p e r t y a a >~ ~ are e x a c t l y those m a t r i c e s w i t h t h e properties j u s t described for t h e e l e m e n t s of G~ e x c e p t for t h e condition on t h e d e t e r m i n a n t .

T h e s y m m e t r i c g r o u p ~ of degree n is considered as a s u b g r o u p of A u t (E) a c t i n g b y p e r m u t a t i o n of t h e x~. T h e g r o u p | is a s u b g r o u p of G:~. W e consider .also t h e direct p r o d u c t ~ = @~,• • | as a s u b g r o u p of | in t h e obvious fashion.

:Clearly ~ N G~ = ~ .

LEMMA 3.11. Let fl and 7 be two norms each of which is canonical with respect to {x~}. I[ I ( f , [ q t ~ ] ) = 1 ( 7 , [qt~]) [or all t, then there is a a e ~ such that y =aft.

Pros/. F o r convenience, we qta(xa+ 0 > 1. T h e n it is e a s y to i > / ~ . Hence, t h e h y p o t h e s i s on {fl(Xl) . . . f(x~l), q-lf(X~i+l ) . . .

set #j = v I + ... + vj. Suppose 0 ~< t ~ 1, a n d qto~(x~,j) <~ 1, v e r i f y t h a t [q~a] (xi) = 1 for i ~</zj a n d [q~a] (xi) = q for fl a n d 7 implies t h a t f o r ~ = 1 . . . g we h a v e q-lf(Xn) } = {7(Xl) . . . 7(Xt,i), q-ly(x~,i+l) . . . . , q-17(Xn) }.

:By equality, we m e a n m u l t i p l i c i t y of repetitions b u t disregarding order.

L e t h be a n y positive integer. W e f o r m t h e s u m of t h e h t h powers of t h e ele- m e n t s of t h e t w o sets j u s t considered. Then, we h a v e

X f(x,)

i = i i > l ~ j =

Assigning to ] successively t h e values 1, 2 . . . g leads to t h e following:

yl vl yl+y~ vl+~2

a n d so on. Since these relations hold for all h>~ 0, we conclude t h a t

{ f ( X l ) , . . . , f ( x , , ) } = . . .

{f(xvl+l) . . . . , fl(x~,+~)} ={7(x~,+1) . . . y(x~,+,~)}, etc. Thus, t h e r e is a n e l e m e n t a e ~ s u c h t h a t af(x~)=7(x~) for all i. Since f is canonical w i t h respect t o {x~}, t h e s a m e is t h e ease f o r aft, a n d since y is also canonical w i t h respect t o {x~}, we conclude finally t h a t 7 = aft.

COROLLARY 3.12. Let f and y be two norms. Then, the ]ollowing statements are equivalent:

(i) There is a v EG~ with Y = v f (ii) I(/~, [ q t ] ) = 1 ( 7 , [q~ ]), /or all t.

160 O. G O L D M A N A N D N . I W A H O R I

Proo/.

(i) ~ (ii).

This implication follows immediately from the definition of the elementary di- visors.

(ii) ~ (i).

B y L e m m a 3.10 we can find elements ~, a , ~ G ~ such t h a t fffl and a~ are canon- ical with respect to {x~}. Since I(~fl, [ q t ] ) = I ( f l , [qta]), with a similar statement for ~, L e m m a 3.11 m a y be applied to give the desired conclusion.

We denote b y D the group of diagonal matrices whose elements are powers of a n d b y (~ the group generated b y D and ~n. Clearly D is a no,Trial subgroup of (~ and (~ is the semi-direct product of D and ~ .

I ) ~ O P O S I T I O ~ 3.13. ~ N { ~ = ~ .

Proo[. Obviously ~ ~ ~ N ( ~ . Suppose &o E (~ N G~ with ~ E D and w E @n. Since 1/q<a(x,)<~ 1, it follows immediately t h a t ~ 1 and hence t h a t w ~ .

T~]:OR]~M 3.14. A u t ( E ) = G ~ ) G ~ .

Proo[. L e t a E A u t ( E ) and set fl = a a . B y l e m m a 3.10, there is a e E G~ such t h a t

~fi = ~ q a is canonical with respect to {x~}. Hence, from the properties of Cm (g), there is a ~ E {~ such t h a t ~Qa~ = ~. B u t this shows t h a t ~Qa E G~ or a E G~ (~ G~.

The theorem just proved asserts t h a t each double coset of A u t (E) with respect to G~ is represented b y an element of (~. We consider now the question of when two elements of ~ represent the same double eoset.

THEOREM 3.15. Suppose that ~ and a are in (~ with ~EG~aG~. Then e E ~ a ~ . Proo/. We h a v e ~ = 2 a # with ~,~uEG~. Hence, ~ = 2 a : r so t h a t I(Qa,[q~]) = I(a~, [qt ]), for all t. Also, because ~ , a E ( ~ , both r a n d a~ are canonical with re- spect to (x~}. I t follows from L e m m a 3.11 t h a t ~ a = ~ a ~ , with ~ E ~ . Hence, a - l T - l ~ ~ ~a, while clearly O~-IT-I~ ~ (~. Since (~ N G~ = ~ , we find t h a t ~ E ~ a ~ .

Section 4. Quadratic forms

F r o m now on we assume t h a t the characteristic of K is not 2. By a non.de- generate quadratic /orm we shall mean a mapping cp:E-->K which is such t h a t

q)(ax) = a 2 qJ(x), and B(x, y) = ~(x + y) - qJ(x) - q~(y)

THE SPACE OF 0-ADIC NORMS 161

is a non-degenerate bilinear f o r m on E. W e shah s a y t h a t ~ is de/inite if ~o(x)= 0 implies x = 0, otherwise ~v is inde/inite. W e denote the orthogonal group of ~0 b y 0 ( ~ ) . T H E O R E M 4.1. Let ~v be a de/inite quadratic /orm. Then 199189 is a norm on E.

Proo/. I t is clear t h a t we need only p r o v e t h a t Iqv(x-4-y)[~< sup (]~o(x) l, I~(Y) I)- Suppose the c o n t r a r y is the case for a p a i r of elements x, y of E, i.e., [ ~ ( x + y ) [ > sup (l~~ I~(Y)I)- T h e n it is clear t h a t x a n d y are linearly independent. Set ~0(z)=a,

~ ( y ) = c a n d B ( x , y ) = b . T h e n we h a v e ]bl>la I a n d also [ b [ > l c ]. W i t h t a n a r b i t r a r y e l e m e n t of K , we h a v e

~ q ~ ( x + ! t y ) = t + t + ~ , 2 ac

(Note t h a t we h a v e t a c i t l y a s s u m e d t h a t e : # 0 . T h a t is so because ~0 is definite.) N o w l ae/b21 < 1, so t h a t the p o l y n o m i a l t 2 + t + ac/b 2 has a simple zero in the field

~ / ~ , hence b y H e n s e l ' s l e m m a it has a zero in K . This contradicts the f a c t t h a t

~0 is definite a n d therefore we conclude t h a t I~[ 89 is indeed a n o r m .

Given % we d e n o t e b y T/(~) t h e set of those n o r m s ~ on E h a v i n g t h e p r o p e r t y [~s(x)[89 all x E E . I t is easy to see t h a t T/(~) is n o t e m p t y . For, let fl be a n y n o r m on E. T h e n [q~(x)189 x # O defines a continuous function on the c o m p a c t space P(E), a n d is therefore bounded. H e n c e t h e r e is some positive real n u m b e r c such t h a t c fl e T/(~v). W e shall also use t h a t n o t a t i o n s for s N ~/(~). I t is clear t h a t b o t h ~/(~s) a n d s are stable u n d e r t h e action of 0 ( ~ ) .

W e s h a h be interested in t h e set of elements of ~(qg)which are m i n i m a l (in t h a t set) in t h e p a r t i a l ordering of ~/. I n order to p r o v e t h a t such m i n i m a l elements exist, it will be useful to e x t e n d t h e notion of n o r m b y defining a semi-norm on E to b e a real-valued function ~ on E such t h a t :

(1) ~(x) >~ 0 (2) ~(ax) =

lal

(3) 7(x + y) ~< sup (~(x), ~(y)).

L e t $ be t h e set of all semi-norms, a n d let

S(~v)={~xE Sllcf(x)189 all x E E } . LEMMA 4.2. S(~) = ~ ( ~ ) .

Proo/. L e t zr be a s e m i - n o r m such t h a t Iq~(x)l~(x) 2 for all x, a n d suppose t h a b

~(y) = 0. I f x is a n y element of E a n d a a n y e l e m e n t of K , we h a v e ~(x + ay) <. a(x). N o w ,