Genus two curves covering elliptic curves: a computational approach

Abstract

A genus 2 curve $C$ has an elliptic subcover if there exists a degree $n$ maximal covering $\psi :C\to E$ to an elliptic curve $E$. Degree $n$ elliptic subcovers occur in pairs $(E,E^{\prime })$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ to the product $E\times E^{\prime }$. We say that $J_C$ is $(n,n)$-split. The locus of $C$, denoted by ${Ł}_n$, is an algebraic subvariety of the moduli space $\M_2$. The space ${Ł}_2$ was studied in Shaska/Völklein and Gaudry/Schost. The space ${Ł}_3$ was studied in Shaska (2004) were an algebraic description was given as sublocus of $\M_2$. In this survey we give a brief description of the spaces ${Ł}_n$ for a general $n$ and then focus on small $n$. We describe some of the computational details which were skipped in Shaska/Völklein and Shaska (2004). Further we explicitly describe the relation between the elliptic subcovers $E$ and $E^{\prime }$. We have implemented most of these relations in computer programs which check easily whether a genus 2 curve has $(2,2)$ or $(3,3)$ split Jacobian. In each case the elliptic subcovers can be explicitly computed.

arXiv admin note: substantial text overlap with arXiv:1209.0439

References in corpus (1)

• Hyperelliptic curves with reduced automorphism group A5