Shimura subvarieties in the Prym locus of ramified Galois coverings

We study Shimura (special) subvarieties in the moduli space $$A_{p,D}$$ A p , D of complex abelian varieties of dimension p and polarization type D. These subvarieties arise from families of covers compatible with a fixed group action on the base curve such that the quotient of the base curve by the group is isomorphic to $${{\mathbb {P}}}^1$$ P 1 . We give a criterion for the image of these families under the Prym map to be a special subvariety and, using computer algebra, obtain 210 Shimura subvarieties contained in the Prym locus.

Shimura curves in the Prym locus

We study Shimura curves of PEL type in [Formula: see text] generically contained in the Prym locus. We study both the unramified Prym locus, obtained using étale double covers, and the ramified Prym locus, corresponding to double covers ramified at two points. In both cases, we consider the family of all double covers compatible with a fixed group action on the base curve. We restrict to the case where the family is one-dimensional and the quotient of the base curve by the group is [Formula: see text]. We give a simple criterion for the image of these families under the Prym map to be a Shimura curve. Using computer algebra we check all the examples obtained in this way up to genus 28. We obtain 43 Shimura curves contained in the unramified Prym locus and 9 families contained in the ramified Prym locus. Most of these curves are not generically contained in the Jacobian locus.