  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.
  This paper is a revised version of ANT-0151, titled `Self-dual ell-adic
representations of finite groups'.
