20 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
Use (5) and equation
(3)
of Section
1
to translate
(1)-(3)
via
f
from
1i
to
H'.
Let
qj
=
f-
1
(Q.;)
E
H'(K),
for j = 1, ...
,m.
Then,
(1')
qj
E
H'(K(t)),
for each
t
E
Xj,
for j =
1, ... ,
m.
We must find p'
E
H'(K)
such that
(2')
p'
E H'(K(a))
for each
a E G(K),
that is, p'
E H'(K);
(3')
liP'-
qjll~ ~
v2
III/II~
in
K(t),
for each
t
E
Xj,
for j =
1, ... ,
m.
Suppose
t
E
Xj
for some 1
~
j
~
m.
As
11/11~
is algebraic over
K,
there
is
aj
E
K
with
a] II/II~
in
K(t).
Replace
xl, ...
'Xm
by a finer partition to
assume this is true for each
t E
Xi.
Thus
(3')
follows from a stronger statement:
(3")
liP'-
qjll~ ~ v
2
la]
in
K(t),
for each
t
E
Xj,
for j =
1, ... ,m.
By Proposition 1.2 there is p' E
H'(K)
such that (3") holds. o
THEOREM
5.2.
Let Ll K be a finite Galois extension with L not formally
real
and
let
rr:
H - G(LI K) be
an
epimorphism of finite groups. For each
1
~
j
~
m
let Ij
~
H
be
a conjugacy
domain of involutions such that rr(Ij)
=
I xi ( L I K). Then there exists
a
regular extension F of L, Galois over K ( x),
and an
isomorphism h1 :G(FIK(x))- H that maps Ix;(FIK(x)) onto Ij· In
addition, the following diagram commutes.
G ( F
I
K (X))
_----'.::h.!..._l
--+
H
r~
/
G(LIK)
In particular,
h1
maps I(FIK(x)) onto
Uj
Ij·
PROOF. By Skolem-Lowenheim Principle
[FJ,
Proposition 6.4] we may as-
sume that K
~
C. We divide the proof into five parts.
Part
1.
Weakening of commutativity.
Let
G
= Ker(
1r).
Instead of commuta-
tivity of the diagram it suffices to show that h1 maps
G(FI L(x))
ont~
G.
Indeed,
apply Lemma
3.3.
This gives an epimorphism of finite groups p:
H - H
and
conjugacy domains of involutions
h, ... ,
fm ~
ii
with
p(f;)
=
Ij.
In addition,
every automorphism of
G(LI K)
that preserves the
Ix; (LI K)
lifts (under p orr)
to an automorphism of
ii
that preserves the
'4.
Let
G
= Ker(p orr).
Assume that we can find a regular extension
P
of
L,
Galois over
K(x),
and an isomorphism
h1
:G(FIK(x)) -
ii
that maps
G(FIL(x))
onto
G
and
the
Ix;(FIK(x))
onto the
fj.
In particular, Ker(rr o po
h1
)
=
G(F/L(x))
=
Ker(resL). Hence there exists an automorphism a of
G(L/ K)
such that a o
1r
o
p
o
h1
=res£ and a preserves the
Ix; (L/ K).
We can lift a to an automorphism
a
of
ii
that preserves the
ij.
Thus, by composing h
1
with
a
we may assume
that (rr o
p)
o
h1
=res£.
Now let F
be
the fixed field of Ker(p) in
P.
Then h1 induces an isomorphism
h
1
:
G(F/ K(x))- H
with the required properties.
Previous Page Next Page