Genus two curves covering elliptic curves: a computational approach

摘要

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 $n$ ellipticsubcovers occur in pairs $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous ofdegree $n^2$ to the product $E \times E'$. We say that $J_C$ is $(n, n)$-split.The locus of $C$, denoted by $\L_n$, is an algebraic subvariety of the modulispace $\M_2$. The space $\L_2$ was studied in Shaska/V\"olklein andGaudry/Schost. The space $\L_3$ was studied in Shaska (2004) were an algebraicdescription was given as sublocus of $\M_2$. In this survey we give a brief description of the spaces $\L_n$ for a general$n$ and then focus on small $n$. We describe some of the computational detailswhich were skipped in Shaska/V\"olklein and Shaska (2004). Further weexplicitly describe the relation between the elliptic subcovers $E$ and $E'$.We have implemented most of these relations in computer programs which checkeasily whether a genus 2 curve has $(2, 2)$ or $(3, 3)$ split Jacobian. In eachcase the elliptic subcovers can be explicitly computed.