Superelliptic curves with many automorphisms and CM Jacobians

Let \mathcal {C} be a smooth, projective, genus g\geq 2 curve, defined over \mathbb {C}. Then \mathcal {C} has many automorphisms if its corresponding moduli point \mathfrak {p} \in \mathcal {M}_g has a neighborhood U in the complex topology, such that all curves corresponding to points in U \setminus \{\mathfrak {p} \} have strictly fewer automorphisms than \mathcal {C}. We compute completely the list of superelliptic curves \mathcal {C} for which the superelliptic automorphism is normal in the automorphism group \mathrm {Aut} (\mathcal {C}) and \mathcal {C} has many automorphisms. For each of these curves, we determine whether its Jacobian has complex multiplication. As a consequence, we prove the converse of Streit’s complex multiplication criterion for these curves.