On Shimura subvarieties generated by families of abelian covers of $\mathbb{P}^{1}$

TL;DR

This paper investigates the presence of Shimura subvarieties generated by families of abelian Galois covers of the projective line within the moduli space of abelian varieties, using computational and characteristic p methods.

Contribution

It combines computational techniques and monodromy analysis to identify and exclude many cases of Shimura subvarieties arising from abelian covers of .

Findings

Identifies conditions under which Shimura subvarieties occur.

Uses characteristic p methods to exclude many cases.

Provides a framework for analyzing Galois covers in moduli spaces.

Abstract

We study the locus of abelian Galois covers of $\mathbb{P}^{1}$ in $A_{g}$ and the problem of occurrence of Shimura (special) subvarieties generated by these covers in the Torelli locus $T_{g}$ inside $A_{g}$. We first investigate the existence of Shimura subvarieties in the mentioned locus by some computational methods based on Moonen-Oort works and then exclude many cases using both characteristic $p$ methods and monodromy computations.