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

A. Silverberg, Yuriy G. Zarkhin

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)25-45
Number of pages21
JournalCompositio Mathematica
Volume126
Issue number1
DOIs
StatePublished - Dec 1 2001

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Abelian Variety
Inertia
Polarization
Subgroup
Isogeny
General Linear Group
Symplectic Group
Positive Characteristic
Euler Characteristic
Divisible
Sheaves
Invertible
Quotient
Finite Group
Odd

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory

Cite this

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

In: Compositio Mathematica, Vol. 126, No. 1, 01.12.2001, p. 25-45.

Research output: Contribution to journalArticle

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