From hyperelliptic to superelliptic curves

TL;DR

This survey extends the theory of elliptic and hyperelliptic curves to superelliptic curves, covering automorphisms, moduli space stratification, invariants, Jacobians, and recent developments.

Contribution

It provides a comprehensive extension of classical curve theories to superelliptic curves, including new results and open problems.

Findings

Analysis of automorphism groups of superelliptic curves

Stratification of the moduli space for superelliptic curves

Recent advances and open problems in superelliptic curve theory

Abstract

In this long survey article we show that the theory of elliptic and hyperelliptic curves can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\mathcal{M}_g$, binary forms, invariants of curves, weighted projective spaces, minimal models for superelliptic curves, field of moduli versus field of definition, theta functions, Jacobian varieties, addition law in the Jacobian, isogenies among Jacobians, etc. Many recent developments on the theory of superelliptic curves are provided as well as many open problems.