It is well known that every finite subgroup of GLd(Qℓ) is conjugate to a subgroup of GLd(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 Sp2d(Qℓ) of inertia type, then G is conjugate in GL2d(Qℓ) to a subgroup of Sp2d(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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory