Decomposed Richelot isogenies of Jacobian varieties of hyperelliptic curves and generalized Howe curves

TL;DR

This paper characterizes decomposed Richelot isogenies between Jacobian varieties of hyperelliptic curves, introduces generalized Howe curves, and explores their relationships, providing new examples of such curves of genus 4 and 5.

Contribution

It offers a comprehensive characterization of decomposed Richelot isogenies for hyperelliptic Jacobians and introduces generalized Howe curves with new explicit examples.

Findings

Characterization of decomposed Richelot isogenies for hyperelliptic Jacobians

Introduction of generalized Howe curves and their relation to Richelot isogenies

New explicit examples of generalized Howe curves of genus 4 and 5

Abstract

We advance previous studies on decomposed Richelot isogenies (Katsura--Takashima (ANTS 2020) and Katsura (ArXiv 2021)) which are useful for analysing superspecial Richelot isogeny graphs in cryptography. We first give a characterization of decomposed Richelot isogenies between Jacobian varieties of hyperelliptic curves of any genus. We then define generalized Howe curves, and present two theorems on their relationships with decomposed Richelot isogenies. We also give new examples including a non-hyperelliptic (resp.\,hyperelliptic) generalized Howe curve of genus 5 (resp.\,of genus 4).