From hyperelliptic to superelliptic curves

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.

survey paper on hyperelliptic and superelliptic curves; 96 pages

References in corpus (4)

• Determining equations of families of cyclic curves
• Semistable types of hyperelliptic curves
• The case for superelliptic curves
• Thomae's Derivative Formulae for Trigonal Curves

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• Moduli relations between l-adic representations and the regular inverse Galois problem
• Superelliptic curves with minimal weighted moduli height
• On the semistability of binary forms over number fields
• Construction of Hyperbolic Signal Sets from the Uniformization of Hyperelliptic Curves