REAL HILBERTIANITY-TOTALLY REAL NUMBERS 17

As p is K-rational,

~(p)

= p. Thus, c(,B(p)) = ,B(p) and ,B(p) is IR.-rational.

Also, as 8t is defined over Q, it commutes with ,B. Therefore (a) is equivalent to

(a') 8t(,B(p)) = c(,B(p)).

Finally, let

F'/IR.(x)

be the Galois extension and

h~:G(F'/IR.(x)) ~

Aut(G)

the embedding corresponding to ,B(p) over JR., and let H' be the image of

h~.

Then by (2.19), condition (d) follows from

(d')

where P' is the unique ordering of JR.. Thus, replacing K by JR. and p by ,B(p),

we may assume that K = JR. and . =

1.

Part IV. K =JR. and.=

1.

By Remark 4.2(d) we have pEN. Write pas

[a, 0, /]. Then a= W'(p) = -q,(p) is IR.-rational. Lemma 4.3 gives assertion (a).

Part V.

Proofof(d).

By (c)-follows from (a)-we have

G1(~:)

$H.

Check:

IGXJ(~:)I

= 2 ·IGI = 2 · [F: C(x)] = [F: IR.(x)] =

IHI.

SoH=

G1(~:).

Write pin the form p =

[x,

h], with

x:

X~

pt (2.1). Fix a pointy E x-t(o).

Let

Yo

= pt,(aU { oo} ), and let 1/J:

Yo

~Yo

be the universal unramified covering

of

Yo.

Fix a point

y

E 1/J-t(O). Put

Y

= x-t(Yo) ;

X.

As

x:

Y

~

Yo

is

unramified, there exists a unique covering ¢:

Yo

~

Y such that

x

o ¢ =

'1/J

and

¢(f))

=

y.

Let

F

be the field of algebraic meromorphic functions on

Yo

(in the

sense of [KN, p. 199]). Then the field extension

F/C(x)

induced by

'1/J

is the

maximal extension of

C(x)

unramified in

Y0

.

Let

F

= C(X) =

C(Y).

We identify

G(F/C(x))

with

G

via

ho,

and

G(F/IR.(x))

with H via ht (see (2.16) and (2.17)). Then, h:Aut(X/lP't)

~

G is the canon-

ical isomorphism Aut(X/lP't)

~

G(F/C(x))

sending

a

E Aut(X/lP't) to the el-

ement

f

t-+

f

oa-t

of

G(F/C(x)).

Similarly, let

G

=

G(F/C(x)),

and let

h:

Aut(Yo/Yo)

~

G

be the canonical map sending

&

to the element

j

t-+

j

o a:-t.

Let

~:lit

(lP't '-.a, 0)

~

Aut(X/lP't) be the epimorphism associated to the point

y E x-t(o) (see (2.2)). Similarly define i: lit

(Y0

,

0)

~

Aut(Y

0

/Y0

),

associated to

the point

f)

E 1/J-t(o). Then there is a commutative diagram

IIt(Yo, 0) ----,,--. Aut(Yo/Yo)

~

G ·

lA• lq)•

h

lresp

lit (IP't ,a, 0) ------.---.- Aut(X/lP't)

~

G

where .* is induced from the inclusion .: Yo

~

pt '-.a.

Put

Oj

=

h

o i('yj), for

j

= 1, ... , r. Then

(9) respai = h o

~

o .*('yi) =

/(''(j)

= aj, for

j

= 1, ... , r.

Let n = (r

+

e)/2 = e

+

m. By Remark 4.2(c) we may assume

at

· · ·

ae

are real, and ae+( 2m+t-j) is the complex conjugate of

ae+j,

for

j

= 1, ... ,

m.