  In this paper we construct Shimura subvarieties of dimension bigger than one
of the moduli space of polarised abelian varieties of a given dimension, which
are generically contained in the Pym loci of (ramified) double covers. The idea
is to adapt the techniques already used to construct Shimura curves in the Prym
loci to the higher dimensional case, namely to use families of Galois covers of
the projective line. The case of abelian covers is treated in detail, since in
this case it is possible to make explicit computations that allow to verify a
sufficient condition for such a family to yield a Shimura subvariety of the
space polarised abelian varieties of a given dimension.
