PRINCIPALLY POLARIZED ABELIAN VARIETIES
7
2.1 THEOREM.
Let (A,
8)
be a principally polarized abelian variety of dimen-
sion g over an algebraically closed field of characteristic p
2::
0,
G :::; Aut(A,
8)
an
abelian subgroup and
1r
the p-part of
I
Aut(A,
8)1.
Then
IGI :::; 7r(39g!)g+l
and
I Aut(A,
8)1 :::;
7r39 (39g!)l6g39
PROOF. 38 is very ample by [Mumford74, 17]. Then by (1.3.1) IAut(A,8)1:::;
1r
39
((38)9) 16939
.
Finally 89 = g! by Riemann-Roch. The estimates now follow
from (1.2.3) and (1.3.1).
D
2.2 COROLLARY.
Under the conditions
of(2.1)
ifp
=
0 orp
(39g!)9+ 1
,
then
1r
=
1, i.e.,
I
Aut(A,
8)1 :::;
(39g!)
16939
PROOF. The order of any
t
E
Aut(A,
8) is at most (39g!)g+l by (1.2.3).
D
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z.
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UNIVERSITY OF WASHINGTON, DEPARTMENT OF MATHEMATICS
Box 354350, SEATTLE, WA 98103
E-mail address:
kovacs@math. washington.
edu
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