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  • Polarizations on Abelian Varieties and Self-dual [ell ]-adic Representations of Inertia Groups

  • A. Silverberg, Yu. G. Zarhin
  • Journal:
    Compositio Mathematica / Volume 126 / Issue 1 / March 2001
    Published online by Cambridge University Press:
    04 December 2007, pp. 25-45
    Print publication:
    March 2001
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  • It is well known that every finite subgroup of GLd(Q[ell ]) is conjugate to a subgroup of GLd(Z[ell ]). 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 [ell ]>d+1, and G is a subgroup of Sp2d(Q[ell ]) of inertia type, then G is conjugate in GL2d(Q[ell ]) to a subgroup of Sp2d(Z[ell ]). We give examples which show that the bound is sharp. We apply these results to construct, for every odd prime [ell ], isogeny classes of Abelian varieties all of whose polarizations have degree divisible by [ell ]2. We prove similar results for Euler characteristic of invertible sheaves on Abelian varieties over fields of positive characteristic.