An explicit expression of the Richelot isogeny through Kleinian hyperellyptic functions

TL;DR

This paper derives an explicit formula for the Richelot isogeny between genus 2 Kummer surfaces using Kleinian hyperelliptic functions, enabling better understanding of their relationships.

Contribution

It provides the first explicit expression of the Richelot isogeny in terms of Kleinian hyperelliptic functions, linking isogenous curves through these functions.

Findings

Explicit Richelot isogeny formula derived

Relation established between Kleinian functions of isogenous curves

Enhances computational approaches to genus 2 curves

Abstract

We find an explicit expression for the Richelot isogeny of Kummer surfaces of genus 2 curves in terms of Kleinian hyperelliptic functions of weight 2. We use this expression to relate Kleinian hyperelliptic functions associated to Richelot isogenous curves.