On Shimura curves generated by families of Galois $G$-covers of curves

Abstract

In this paper we prove there are no families of cyclic $\Z_n$-covers of elliptic curves which generate non-compact Shimura (special) curves that lie generically in the Torelli locus $T_g$ of abelian varieties with $g\geq 8$ when $n$ has a proper prime factor $p\geq 7$. This non-existence is also shown for families of $\Z_n$-covers of curves of any genus $s$ and when $n$ has a large enough prime number $p$ (depending on $s$). We achieve these results by applying the theory of Higgs bundles and the Viehweg-Zuo characterization of Shimura curves in the moduli space of principally polarized abelian varieties.