### Abstract

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 ℓ^{2}. We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.

Original language | English (US) |
---|---|

Pages (from-to) | 25-45 |

Number of pages | 21 |

Journal | Compositio Mathematica |

Volume | 126 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 2001 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

### Cite this

*Compositio Mathematica*,

*126*(1), 25-45. https://doi.org/10.1023/A:1017508716337

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*Compositio Mathematica*, vol. 126, no. 1, pp. 25-45. https://doi.org/10.1023/A:1017508716337

**Polarizations on Abelian Varieties and Self-dual ℓ-adic Representations of Inertia Groups.** / Silverberg, A.; Zarkhin, Yuriy G.

Research output: Contribution to journal › Article

TY - JOUR

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

AU - Silverberg, A.

AU - Zarkhin, Yuriy G.

PY - 2001/12/1

Y1 - 2001/12/1

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.

UR - http://www.scopus.com/inward/record.url?scp=0041328193&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041328193&partnerID=8YFLogxK

U2 - 10.1023/A:1017508716337

DO - 10.1023/A:1017508716337

M3 - Article

VL - 126

SP - 25

EP - 45

JO - Compositio Mathematica

JF - Compositio Mathematica

SN - 0010-437X

IS - 1

ER -