18 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
Observe that
F
is the maximal extension of
JR(x)
unramified outside the primes
of
IR(x)
induced by ai, ... , ar,
oo.
In this situation the proof of
[KN,
Satz
2]
shows that there is f.
E
fi
=
a(F jlR(x))
such that €, cTI, ... ,
cTn
form a system
of generators for
fi
with the defining relations
(10)
€2
=
1
and
for 1 :::; j :::;
e.
Further, u!+j = a-;~(
2
m+I-j) for j = 1, ... , 2m. By (3) and (4) this implies that
resp€ and
f.
act on
a
in the same way. Since H is a subgroup of Aut(a), this
implies that resp€
=E.
Each involution of
fi
is conjugate to one of €, cTI €, u2ui €, ... ,
u
e · · ·
u2ui €.
Indeed, by
[HJ2,
Lemma 4.2 (Part E)],
fi
is the free profinite product of the
free profinite group (ue+I, ... ,
un)
of rank
n- e
=
m
withe+ 1 groups
(€), (a-If.), (a-2a-I€), ... , (a-e · · · u2ui€)
that are of order two. Thus by
[HR,
Theorem A'] the elements of finite order in
fi
are the conjugates of the elements of these e + 1 subgroups.
By Lemma
4.5
below, all involutions of
fi
are real. Using (9) and
(5)
conclude
I(F/IR(x))
=
respl(F
/IR(x))
=
respConif( {€, cTIE, ... ,
cTe · · ·
ui€})
=
=
ConH( {
E, O"If., ... ,
O"e
O"If.})
=
ConH(I).
o
LEMMA
4.5.
Let S be
a
finite set of finite prime divisors of the field
JR(x).
Let IR(x)s be the maximal extension of!R(x) unramified outsideS
U {oo},
and
set as= a(JR(x)s jlR(x)). Then all involutions of as
are
real.
PROOF. By
[KN,
Satz
3]
the absolute Galois group
g
of
IR(x)
has generators
{o,
Tp
I
p
a finite prime of
JR(x)/IR}
with the defining profinite relations
82 =
1 and r:
= (
II
r;~)
r;I (
II
r;~)
-I for all real
p.
p 'p p 'p
Here
TIP
'p r;~ is the unique accumulation point of the products r;~ · · · r;~ E
g
for real primes PI, ... , Pr with PI
· · ·
Pr
p.
Furthermore,
[KN]
constructs this system of generators in such a way that for every finite set
S
of
finite primes and every finite prime
p
f/.
S
the natural restriction map
g
~
as
maps
Tp
onto
1 [KN,
p.
207].
Let PI
· · ·
Pe be the real, and Pe+I, ... , Pn the complex primes of
S.
Let
cTj
be the image of
Tp
j,
for j
=
1, ... ,
n,
and let E be the image of
0
in
as.
Then
€, cTI, ... ,
cTn
generate
as
and satisfy (10). These are in fact defining relations
for
as
by
[KN,
Satz
2].
As in the last part of the proof of Proposition 4.4,
each involution of
as
is conjugate to some
cTj · ·
ui€, where 0:::; j :::;
e.
Thus it
suffices to show that each
cTj · · ·
cTIE lifts to an involution of
9.
To this end put
Oo=O and Oj=(
II
r;~)-Io=rp 1 (
II
r;~)-Io
p
'~p j
p 'p
j
Previous Page Next Page