Genus two curves covering elliptic curves: a computational approach

TL;DR

This paper explores the algebraic and computational aspects of genus 2 curves with elliptic subcovers, providing explicit descriptions and algorithms for identifying and computing such structures.

Contribution

It offers a detailed description of the spaces of genus 2 curves with elliptic subcovers for small degrees and implements algorithms for their explicit computation.

Findings

Explicit algebraic descriptions of $ extstyle ext{L}_n$ for small n.

Algorithms for checking if a genus 2 curve has split Jacobian.

Computational methods for explicitly finding elliptic subcovers.

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 $(E, E')$. The Jacobian $J_C$ of $C$ is isogenous of degree $n^2$ 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 moduli space $\M_2$. The space $\L_2$ was studied in Shaska/V\"olklein and Gaudry/Schost. The space $\L_3$ 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 $\L_n$ for a general $n$ and then focus on small $n$. We describe some of the computational details which were skipped in Shaska/V\"olklein and Shaska (2004). Further we explicitly describe the relation between the elliptic subcovers $E$ and $E'$. 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.