TY - JOUR

T1 - Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups

AU - Silverberg, A.

AU - Zarhin, Yu G.

N1 - Funding Information:
Silverberg would like to thank NSA, NSF, the Science Scholars Fellowship Program at the Bunting Institute, UC Berkeley, and MSRI; Zarhin would like to thank NSF.

PY - 2001

Y1 - 2001

N2 - 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.

AB - 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.

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U2 - 10.1023/A:1017508716337

DO - 10.1023/A:1017508716337

M3 - Article

AN - SCOPUS:0041328193

SN - 0010-437X

VL - 126

SP - 25

EP - 45

JO - Compositio Mathematica

JF - Compositio Mathematica

IS - 1

ER -