Superelliptic curves with many automorphisms and CM Jacobians

TL;DR

This paper classifies superelliptic curves with many automorphisms and determines which have Jacobians with complex multiplication, proving a converse to Streit's criterion for these cases.

Contribution

It provides a complete classification of superelliptic curves with many automorphisms and characterizes their Jacobians with complex multiplication.

Findings

Complete list of superelliptic curves with many automorphisms

Identification of Jacobians with complex multiplication among these curves

Proof of the converse of Streit's CM criterion for these curves

Abstract

Let $\mathcal{C}$ be a smooth, projective, genus $g\geq 2$ curve, defined over $\mathbb{C}$. Then $\mathcal{C}$ has \emph{many automorphisms} if its corresponding moduli point $p \in \mathcal{M}_g$ has a neighborhood $U$ in the complex topology, such that all curves corresponding to points in $U \setminus \{p \}$ have strictly fewer automorphisms than $\mathcal{C}$. We compute completely the list of superelliptic curves having 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.