Self-dual ell-adic representations of finite 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.
These results hold for arbitrary finite groups, not necessarily of inertia 
type. The same statement holds true for orthogonal groups. 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.