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

Higher dimensional Shimura varieties in the Prym loci of ramified double covers