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

In this paper, we construct Shimura subvarieties of dimension bigger than one of the moduli space Ap delta${\mathsf {A}}<^>\delta _{p}$ of delta-polarized abelian varieties of dimension p, which are generically contained in the Prym 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 P1${\mathbb {P}}<^>1$. 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 Ap delta${\mathsf {A}}<^>\delta _{p}$.

Automated summary

In this paper, we construct Shimura subvarieties of dimension bigger than one in the moduli space of delta-polarized abelian varieties of dimension p. We adapt the techniques used to construct Shimura curves to the higher dimensional case by using families of Galois covers of P1. The case of abelian covers is treated in detail, and explicit computations are used to verify the condition for the family to yield a Shimura subvariety of Ap delta${\mathsf {A}}<^>\delta _{p}$.