On the Existence of Shimura curves in the Prym locus of abelian covers of projective line

TL;DR

This paper proves the non-existence of non-compact Shimura curves in the Prym locus of certain abelian covers of the projective line using Higgs bundle theory and the Viehweg-Zuo characterization.

Contribution

It establishes new non-existence results for Shimura curves in the Prym locus of specific abelian covers, extending the understanding of the structure of these loci.

Findings

No non-compact Shimura curves in the Prym locus for certain covers

Utilizes Higgs bundle stability and Viehweg-Zuo characterization

Results apply to covers with group structures $ ext{Z}_{2p}$ and $ ext{Z}_{2p} imes ( ext{Z}_p)^{m-1}$

Abstract

Using the theory of Higgs bundles and their stabitlity properties associated to fibered surfaces and the Viehweg-Zuo characterization of Shimura curves in the moduli space of abelian varieties in terms of Higgs bundles, we prove that there does not exist any non-compact Shimura curves in the Prym locus of totally ramified $\Z_{2p}$- or $\Z_{2p}\times (\Z_{p})^{m-1}$-covers of the projective line in $A_{g}$ for $g\geq 8$, where $p\geq 5$ is a prime number.