Families of Explicit Isogenies of Hyperelliptic Jacobians

TL;DR

This paper constructs explicit families of hyperelliptic curves with isogenies between their Jacobians, demonstrating generic simplicity and describing kernels, based on polynomial classification work.

Contribution

It introduces new explicit families of hyperelliptic Jacobians with isogenies, expanding understanding of their structure and kernels.

Findings

Jacobian families of various genera with explicit isogenies

Jacobian generic absolute simplicity in these families

Explicit description of isogeny kernels

Abstract

We construct three-dimensional families of hyperelliptic curves of genus 6, 12, and 14, two-dimensional families of hyperelliptic curves of genus 3, 6, 7, 10, 20, and 30, and one-dimensional families of hyperelliptic curves of genus 5, 10 and 15, all of which are equipped with an an explicit isogeny from their Jacobian to another hyperelliptic Jacobian. We show that the Jacobians are generically absolutely simple, and describe the kernels of the isogenies. The families are derived from Cassou--Nogu\`es and Couveignes' explicit classification of pairs $(f,g)$ of polynomials such that $f(x_1) - g(x_2)$ is reducible.