Special subvarieties arising from families of cyclic covers of the projective line

TL;DR

This paper classifies special subvarieties in the moduli space of abelian varieties arising from families of cyclic covers of the projective line, identifying exactly twenty such families using advanced techniques in mixed characteristics.

Contribution

It provides a complete classification of twenty specific families of cyclic covers that produce special subvarieties in the moduli space, employing Dwork and Ogus's methods.

Findings

Exactly twenty families produce special subvarieties.

The classification uses techniques in mixed characteristics.

The result advances understanding of the structure of moduli spaces.

Abstract

We consider families of cyclic covers of the projective line, where we fix the covering group and the local monodromies and we vary the branch points. We prove that there are precisely twenty such families that give rise to a special subvariety in the moduli space of abelian varieties. Our proof uses techniques in mixed characteristics due to Dwork and Ogus.