16

MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN

Recall that (cf)(cr) =

f("Y)·

Combine (6) and (7) with (3) and (4) to get

cf("Yj)

=

f(crj)

=

aj =(to

f)("Yj), for each 1 :::; j:::;

r.

o

PROPOSITION

4.4. Let (K, P) be an ordered field and tan involution in G(K)

inducing P on K. Assume K

~ C.

Let

A

E

G(Q)

and p

E

1-l(K) with

(8) liP- A(q)llt 2 v 2 in K(t),

and

p

= A(p) is K-rational. Let h1:G(FIK(x))--+ Aut(G) be the embedding

corresponding top over K with Fl K(x) Galois as given in

(2.17).

Put L = K(p)

and let H be the image of

h1

.

The following hold:

(a) 8,(p) = t(p);

(b) P does not extend to L; in particular, P does not extend to F;

(c)

GX1(t) :::; H, and therefore

I~

H;

(d) h1(Ip(FIK(x)))

=

ConH(I).

PROOF.

By (2.14), 8,(p)

'I

p.

Therefore (a) implies t(p)

'I

p.

Hence

L

~

K(t), and this implies (b). Furthermore, the criterion of (2.17) implies that

t E

H.

Since G:::;

H,

GXI(t):::;

H.

So it suffices to prove (a) and (d).

Part

I. Reduction to K with archimedian orderings dense in X(K). Let K

0

be a finitely generated subfield of

K,

containing the finitely generated subfield

Q(p)

of K. Let Po be the restriction of P to K

0

.

This ordering is induced

from the restriction to

E

G ( K

o) of

t.

Let

F0

I

K

0

(

x)

be the Galois extension and

(hi)o: G(Fol Ko(x)) --+Aut( G) the embedding corresponding top over K

0

.

We

may assume that F = Fo · K from (2.18). If K

0

is sufficiently large, then the

restriction map resp0

:

G(F I K(x)) --+ G(Fol Ko(x)) is an isomorphism. If we can

show that the assertions hold for Ko, Po, to, (hi)o, then, by (2.19) and since

to(P) = t(p), they also hold for

K, P,

t,

h1. Lemma 1.6(a) shows the set of

archimedian orderings on K

0

is dense in X(K0

).

So we may assume that K

enjoys this property.

Part

II. Reduction toP archimedian. By Remark 1.8, if P' is an (archime-

dian) ordering of K sufficiently near to P, then lp(FI K(x)) = Jp,(FI K(x)).

We may assume an involution

t1

E

G(K), so near

tot,

induces

P'

that

liP- A(q)llt 2 =liP- A(q)llt' 2 and t(p) = t'(p).

Thus we may replace

P

by

P'

and

t

by

t

1

•

Part

III. Reduction to K = lR and

A

= 1. Assume that P is archimedian.

Extend A-

1

to an automorphism f3 of C, and let t

1

= /3t/3-

1.

Then f3(K(t)) =

(f3(K))(t') is a real closure of (f3(K),/3(P)). Hence it is also archimedian. Thus

we may assume that (f3(K))(t')

~ JR.

Hence /3t/3-

1

= t

1

= resf3(.K)c, where cis

complex conjugation on C.

Since

q

is algebraic over

Q,

/3A(q) =

q.

So, application of f3 to (8) yields

(8')