10
MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN
phisms of
x
are defined over L [FV1, Cor. I]. Thus, there is a unique cover
XL: XL
----+
Pl
such that base change with the embedding L
----+
C gives
x
from
XL and the automorphisms of
x
from the automorphisms of XL.
(2.16} Fields of definition of automorphisms. We recall some facts from
[FV1, §6.3]. The function field F
=
L(XL) is regular over L, and the exten-
sion F I L(x) induced by
x
is Galois. Here, x is the identity function on JP1
.
The group G(FIL(x)) (acting from the left on F) is canonically isomorphic to
Aut(XIJP1
),
via the map that sends
a
E
Aut(XIJP1
)
to the element
g
t-t
go a-
1
of G(FIL(x)). Let h0
:
G(FIL(x))----+ G be the composition of this isomorphism
with h: Aut(XIJP1)
----+
G.
(2.17} Indentification of automorphisms of G. Furthermore,
Ll
K and
FIK(x) are Galois extensions, and the centralizer of G(FIL(x)) in G(FIK(x))
is trivial. This implies ho extends to a unique embedding h
1:
G(FI K(x))
----+
Aut(G). [FV1, Proposition 3] says:
H
:=
h1(G(FIK(x))) equals
{A
E
Aut(G)I8A(P) is conjugate top under G(LIK)}.
(2.18} Action by autmorphisms of C. Let {3 be an automorphism of C,
and let K and K' be two subfields ofC such that [3(K)
~
K'. Put p'
=
{J(p) and
L'
=
K'(p'). Then [3(L)
~
L', and A(p')
E
if.(K'). Let F'IL'(x) be the Galois
extension associated to K' and the point p' of 1i, and let
h~:
G(F' I K'(x))
----+
Aut( G) be the associated embedding. Then the following holds:
Let {3: L(x)----+ L'(x) be the extension of {3 (fixing x). This map extends further
to {3: F
----+
F' such that canonically
(I)
F'
9:!
{3(F) !59f3(L) L'
9:!
F !59L L'.
Consider restriction {3*: G(F' I K'(x))
----+
G(FI K(x)): u
E
G(F' I K'(x)) goes
to {3-
1uif3(F)f3·
It is injective and it gives an isomorphism G(F'IL'(x))
----+
G(FIL(x)). Further, it makes the following diagram commutative:
/3*
G(F'IK'(x)) -----'--- G(FIK(x))
(2)
~~
Aut(G)
PROOF (2) COMMUTES. We have p'
=
[f3(x),
h
o
{3;
1]
by (2.10). The natural
action of {3
E
Aut(C) on functions defined over L extends {3 to a map from
F
=
L(X) to F'
=
L([J(X)). Then (I) follows from the fact that F is regular over
L, and [F': L'(x)] = [F: L(x)] (= deg(x)). The proof of {2) is straightforward
from the definitions. o
(2.19} Conclusion from {2.18}. In {2.I8) and Lemma 1.9 we have
h~(Jp,(F'IE')) ~
h1(lp(FIE)) and
ConHh~(Jp,(F'IE'))
=
h1(lp(FIE)),
where
His
the image of h1 in Aut{ G).
If
the 'restriction' map {3*: G(F' I K'(x))----+
G(FIK(x)) is an isomorphism, then
h~(Jp,(F'IE'))
=
h1(lp(FIE)). Indeed,
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