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.
Categories: math.AG