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

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 ${\mathbb{Z}}_{2p}$- or ${\mathbb{Z}}_{2p}\times ({\mathbb{Z}}_p{)}^{m-1}$-covers of the projective line in $A_g$ for $g\ge 8$, where $p\ge 5$ is a prime number.

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