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

TL;DR

This paper constructs higher-dimensional Shimura subvarieties within Prym loci of ramified double covers, extending previous methods used for curves to higher dimensions via families of Galois covers.

Contribution

It adapts techniques for constructing Shimura curves to higher dimensions, providing explicit conditions for abelian covers to generate Shimura subvarieties in moduli spaces.

Findings

Constructed higher-dimensional Shimura subvarieties in Prym loci.

Explicit computations for abelian covers verify conditions for Shimura subvarieties.

Extended the framework from curves to higher-dimensional cases.

Abstract

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.