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.