Superelliptic curves with many automorphisms and CM Jacobians

Abstract:

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.