Computation of genus 2 Kleinian hyperelliptic functions via Richelot isogenies

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.