From hyperelliptic to superelliptic curves

Malmendier, Andreas and Shaska, Tony (2019) From hyperelliptic to superelliptic curves. Albanian Journal of Mathematics, 13 (1). pp. 107-200. ISSN 1930-1235

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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. We show that this theory can be extended naturally to all superelliptic curves. We focus on automorphism groups, stratification of the moduli space $\M_g$, 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.

Item Type: Article
Subjects: Q Science >
Divisions: Engineering, Science and Mathematics > Mathematics >
Engineering, Science and Mathematics
Depositing User: Editor Risat
Date Deposited: 31 May 2019 22:33
Last Modified: 31 May 2019 22:33

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