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Abstract

Abstract. It is well known that every ¢nite subgroup of GL d Q is conjugate to a subgroup of GL d Z . However, this does not remain true if we replace general linear groups by symplectic groups. We say that G is a group of inertia type of G is a ¢nite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if b d 1, and G is a subgroup of Sp 2d Q of inertia type, then G is conjugate in GL 2d Q to a subgroup of Sp 2d Z . We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime , isogeny classes of Abelian varieties all of whose polarizations have degree divisible by 2 . We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over ¢elds of positive characteristic.