Isogenous components of Jacobian surfaces

Abstract

Let $\mathcal{X}$ be a genus 2 curve defined over a field $K$, $\mbox{char} K = p \geq 0$, and $\mbox{Jac} (\mathcal X, ι)$ 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.