Computation of genus 2 Kleinian hyperelliptic functions via Richelot isogenies

TL;DR

This paper introduces a numerical algorithm for evaluating genus 2 Kleinian hyperelliptic functions using Richelot isogenies, involving a sequence of isogenous curves and a convergence-guaranteeing selection method.

Contribution

It presents a novel algorithm that computes hyperelliptic functions via Richelot isogenies with a new method for selecting curves to ensure convergence.

Findings

Algorithm successfully evaluates Kleinian functions for genus 2 curves.

Method guarantees convergence of iterative calculations.

Provides a practical approach for hyperelliptic function computation.

Abstract

In this work we propose an algorithm that numerically evaluates Kleinian hyperelliptic functions associated with a complex curve of genus 2. This algorithm is based upon constructing a sequence of curves with Richelot isogenous Jacobians and a recurrent procedures that reduces the calculation to a degenerate curve. As a part of mentioned algorithm we propose a method of choosing a Richelot isogenous curve (among 15 possibilities) that guarantees convergence of the equations of the curves and associated Kleinian functions of weight 2 under iterations.