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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[Iitaka82] S. litaka, Algebraic Geometry, Springer, 1982.

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UNIVERSITY OF WASHINGTON, DEPARTMENT OF MATHEMATICS

Box 354350, SEATTLE, WA 98103

E-mail address:

kovacs@math. washington.

edu