Solution of the Jacobi inversion problem on non-hyperelliptic curves

TL;DR

This paper introduces a method to solve the Jacobi inversion problem for non-hyperelliptic curves using Kleinian $ ext{wp}$ functions, based on multivariable sigma functions, providing explicit solutions for certain algebraic curves.

Contribution

It presents a comprehensive solution to the Jacobi inversion problem on plane algebraic curves via multivariable sigma functions, extending the theory to non-hyperelliptic cases.

Findings

Explicit solutions for trigonal, tetragonal, and pentagonal curves.

Complete resolution of the Jacobi inversion problem on plane algebraic curves.

Application of multivariable sigma functions to non-hyperelliptic curves.

Abstract

In this paper we propose a method of solving the Jacobi inversion problem in terms of multiply periodic $\wp$ functions, also called Kleinian $\wp$ functions. This result is based on the recently developed theory of multivariable sigma functions for $(n,s)$-curves. Considering $(n,s)$-curves as canonical representatives in the corresponding classes of bi-rationally equivalent plane algebraic curves, we claim that the Jacobi inversion problem on plane algebraic curves is solved completely. Explicit solutions on trigonal, tetragonal and pentagonal curves are given as an illustration.