Families of Prym varieties of abelian coverings and Shimura varieties

TL;DR

This paper investigates the structure of special subvarieties in the moduli space of abelian varieties arising from families of abelian covers of the projective line, under the injectivity condition of the Prym map in characteristic p.

Contribution

It establishes that such subvarieties are highly restricted and can only be constructed via the group action of the family when certain conditions are met.

Findings

Special subvarieties are highly restrictive under the Prym map injectivity condition.

Shimura varieties from these families are generated solely by the group action.

Results apply to families with one-dimensionality or specific eigenspace conditions.

Abstract

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