Shimura subvarieties in the Prym locus of ramified Galois coverings

TL;DR

This paper investigates special Shimura subvarieties within the Prym locus of ramified Galois coverings, providing criteria for their identification and discovering 210 such subvarieties through computational methods.

Contribution

It introduces a criterion for identifying Shimura subvarieties in the Prym locus and computationally finds 210 examples, advancing understanding of the structure of these loci.

Findings

Established a criterion for Shimura subvarieties in the Prym locus.

Identified 210 Shimura subvarieties using computer algebra.

Enhanced understanding of the Prym locus structure.

Abstract

We study Shimura (special) subvarieties in the moduli space $A_{p,D}$ 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 $\mathbb{P}^1$. 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.