Curves of genus 2 with (n, n)-decomposable jacobians

Abstract

Let $C$ be a curve of genus 2 and $ψ_1:C \lar E_1$ a map of degree $n$, from $C$ to an elliptic curve $E_1$, both curves defined over $\bC$. This map induces a degree $n$ map $ϕ_1:\bP^1 \lar \bP^1$ which we call a Frey-Kani covering. We determine all possible ramifications for $ϕ_1$. If $ψ_1:C \lar E_1$ is maximal then there exists a maximal map $ψ_2:C\lar E_2$, of degree $n$, to some elliptic curve $E_2$ such that there is an isogeny of degree $n^2$ from the Jacobian $J_C$ to $E_1 \times E_2$. We say that $J_C$ is $(n,n)$-decomposable. If the degree $n$ is odd the pair $(ψ_2, E_2)$ is canonically determined. For $n=3, 5$, and 7, we give arithmetic examples of curves whose Jacobians are $(n,n)$-decomposable.