# Isogenous components of Jacobian surfaces

## TL;DR

This paper classifies certain genus 2 curves with reducible Jacobians having elliptic components that are isogenous, and explores their geometric structures and cryptographic applications.

## Contribution

It proves finiteness results for genus 2 curves with specific isogeny conditions and determines associated Kummer and Shioda-Inose surfaces.

## Key findings

- Finiteness of genus 2 curves with isogenous elliptic components under specified conditions.
- Explicit descriptions of Kummer and Shioda-Inose surfaces for these Jacobians.
- Potential cryptographic applications in positive characteristic fields.

## Abstract

Let $\mathcal X$ be a genus 2 curve defined over a field $K$, $\mbox{char} K = p \geq 0$, and $\mbox{Jac} (\mathcal X, \iota)$ its Jacobian, where $\iota$ is the principal polarization of $\mbox{Jac} (\mathcal X)$ attached to $\mathcal X$. Assume that $\mbox{Jac} (\mathcal X)$ is $(n, n)$- geometrically reducible with $E_1$ and $E_2$ its elliptic components. We prove that there are only finitely many curves $\mathcal X$ (up to isomorphism) defined over $K$ such that $E_1$ and $E_2$ are $N$-isogenous for $n=2$ and $N=2,3, 5, 7$ with $\mbox{Aut} (\mbox{Jac} \mathcal X )\cong V_4$ or $n = 2$, $N = 3,5, 7$ with $\mbox{Aut} (\mbox{Jac} \mathcal X ) \cong D_4$. The same holds if $n=3$ and $N=5$. Furthermore, we determine the Kummer and the Shioda-Inose surfaces for the above $\mbox{Jac} \mathcal X$ and show how such results in positive characteristic $p>2$ suggest nice applications in cryptography.