Special Points Arising From Faithful Metacyclic and Dicyclic Galois Covers of the Projective Line

Abstract

Within the Schottky problem, the study of special subvarieties of the Torelli locus has long been of great interest. We describe a representation-theoretic criterion for a Jacobian variety arising from a $G$-Galois cover of $\mathbb{P}^1$ branched at $3$ points to have complex multiplication (CM). For $G$ faithful metacyclic or dicyclic, we classify all such covers with Galois group $G$, identifying those that have CM. We compute the CM-field and type of Jacobian varieties arising from these covers, applying the representation theory of $G$ over $\mathbb{Q}$ and $\mathbb{Q}(\zeta_4)$. In particular, symplectic irreducible representations of $G$ are afforded by the Jacobian variety in the dicyclic case, giving rise to new examples of CM abelian varieties.