FROM HYPERELLIPTIC TO SUPERELLIPTIC CURVES

Source abstract

The theory of elliptic and hyperelliptic curves has been of crucial importance in the development of algebraic geometry. Almost all fundamental ideas were first obtained and generalized from computations and constructions carried out for elliptic or hyperelliptic curves. In this long survey, we show that this theory can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space ℳg, 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.