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Under the condition that the Prym map is injective in characteristic $p$, weprove that the special subvarieties in the moduli space of abelian varieties ofdimension $l$ and polarization type $D$, $A_{l,D}$, arising from families ofabelian covers of $\P^1$ are of a very restrictive nature. In other words, ifthe family is one-dimensional or if it contains an eigenspace of certain typefor the group action on the cohomology of fibers, then the Shimura varietiesarising from such families can only be constructed by the group action of thefamily.

Families of Prym varieties of abelian coverings and Shimura varieties