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,