Abelian function fields on Jacobian varieties

TL;DR

This paper explores the algebraic structure of Abelian function fields on Jacobian varieties, detailing their properties, addition laws, and applications to algebraic curves, with explicit examples for hyperelliptic and non-hyperelliptic cases.

Contribution

It provides a comprehensive exposition of multiply periodic function fields on Jacobian varieties, including their algebraic models and dynamical applications, with detailed examples.

Findings

Explicit descriptions of Abelian function fields on Jacobian varieties.

Demonstrations of addition laws and algebraic models.

Examples for hyperelliptic and non-hyperelliptic curves.

Abstract

In this paper the fields of multiply periodic, or Kleinian $\wp$-functions are exposed. Such a field arises on the Jacobian variety of an algebraic curve, and provides natural algebraic models of the Jacobian and Kummer varieties, possesses the addition law, and accommodates dynamical equations with solutions. All this will be explained in detail for plane algebraic curves in their canonical forms. Example of hyperelliptic and non-hyperelliptic curves are presented.