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.
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