Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups

It is well known that every finite subgroup of GL_d(Qℓ) is conjugate to a subgroup of GL_d(Zℓ). 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 finite group which has a normal Sylow-p-subgroup with cyclic quotient. We show that if ℓ > d + 1, and G is a subgroup of Sp_2d(Qℓ) of inertia type, then G is conjugate in GL_2d(Qℓ) to a subgroup of Sp_2d(Zℓ). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime ℓ, isogeny classes of Abelian varieties all of whose polarizations have degree divisible by ℓ². We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.