On superelliptic curves of level $n$ and their quotients, I

TL;DR

This paper investigates families of superelliptic curves with fixed automorphism groups, exploring their invariants, algebraic relations, and the structure of their moduli spaces, along with computational tools for their analysis.

Contribution

It introduces a framework for analyzing superelliptic curves via invariants, describes algebraic relations among these invariants, and provides a Maple package for computing normal forms and invariants.

Findings

Algebraic relations among invariants determine inclusion relations among loci.

A Maple package for computing normal forms and invariants is provided.

A complete classification of superelliptic curves of genus ≤ 10 over fields of characteristic ≠ 2 is available in subsequent work.

Abstract

We study families of superelliptic curves with fixed automorphism groups. Such families are parametrized with invariants expressed in terms of the coefficients of the curves. Algebraic relations among such invariants determine the lattice of inclusions among the loci of superelliptic curves and their field of moduli. We give a Maple package of how to compute the normal form of an superelliptic curve and its invariants. A complete list of all superelliptic curves of genus $g \leq 10$ defined over any field of characteristic $\neq 2$ is given in a subsequent paper.