Computation of $\wp$-functions on plane algebraic curves

TL;DR

This paper introduces numerical tools and analytical methods for computing Kleinian $ ext{wp}$-functions on plane algebraic curves, utilizing explicit radical solutions and Riemann surface models for hyperelliptic, trigonal, and tetragonal curves.

Contribution

It presents a novel analytical approach to constructing Riemann surfaces and computing $ ext{wp}$-functions directly from divisors, improving control and verification methods.

Findings

Explicit radical solutions enable direct computation of $ ext{wp}$-functions.

The approach applies to hyperelliptic, trigonal, and tetragonal curves.

A method for identifying the Riemann constants characteristic is proposed.

Abstract

Numerical tools for computation of $\wp$-functions, also known as Kleinian, or multiply periodic, are proposed. In this connection, computation of periods of the both first and second kinds is reconsidered. An analytical approach to constructing Riemann surfaces of plane algebraic curves of low gonalities is used. The approach is based on explicit radical solutions to quadratic, cubic, and quartic equations, which serve for hyperelliptic, trigonal, and tetragonal curves, respectively. The proposed analytical models of Riemann surfaces give full control over computation of the Abel image of any point or divisor. Therefore, computation of $\wp$-functions at Abel images of given divisors can be done directly. An alternative computation with the help of the Jacobi inversion problem is used for verification. Hyperelliptic and trigonal curves are considered in detail, and illustrated by examples. A method of finding the unique characteristic corresponding to the vector of Riemann constants is suggested for non-hyperelliptic and hyperelliptic curves.