  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.
