### Polarizations on abelian varieties and self-dual ell-adic representations of inertia groups, by A. Silverberg, Yu. G. Zarhin

It is well-known that every finite subgroup of GL_d(Q_{\ell}) is
conjugate to a subgroup of GL_d(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 if 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 Sp_{2d}(Q_{\ell}) of inertia type,
then G is conjugate in GL_{2d}(Q_{\ell}) to a subgroup of
\Sp_{2d}(Z_{\ell}). Despite the fact that G can fail to be conjugate
in \GL_{2d}(Q_\ell) to a subgroup of \Sp_{2d}(Z_\ell), we prove that
it can nevertheless be embedded in \Sp_{2d}(F_\ell) in such a way
that the characteristic polynomials are preserved (mod \ell), as long
as \ell>3. The latter result holds for arbitrary finite groups, not
necessarily of inertia type, and holds also for symmetric forms, not
just alternating forms. We give examples which show that the bounds are
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.

A. Silverberg, Yu. G. Zarhin <silver@math.ohio-state.edu>