Shimura varieties and abelian covers of the line

TL;DR

This paper investigates conditions under which families of abelian covers of the projective line do not form higher dimensional Shimura subvarieties in the moduli space of abelian varieties, using monodromy and reduction to p arguments.

Contribution

It provides new criteria and methods to identify when such families do not produce special subvarieties, advancing the classification of special families in the moduli space.

Findings

Families of abelian covers generally have large monodromy groups.

Under certain conditions, these families do not give rise to Shimura subvarieties.

Two-dimensional subvarieties in this locus are not special.

Abstract

We prove that under some conditions on the monodromy, families of abelian covers of the projective line do not give rise to (higher dimensional) Shimura subvarieties in $A_g$. This is achieved by a reduction to $p$ argument. We also use another method based on monodromy computations to show that two dimensional subvarieties in the above locus are not special. In particular it is shown that such families have usually large monodromy groups. Together with our earlier results, the above mentioned results contribute to classifying the special families in the moduli space of abelian varieties and partially completes the work of several authors including the author's previous work.