Families of genus two curves with many elliptic subcovers

Abstract

We determine all genus 2 curves, defined over $\mathbb{C}$, which have simultaneously degree 2 and 3 elliptic subcovers. The locus of such curves has three irreducible 1-dimensional genus zero components in ${\mathcal{M}}_2$. For each component we find a rational parametrization and construct the equation of the corresponding genus 2 curve and its elliptic subcovers in terms of the parameterization. Such families of genus 2 curves are determined for the first time. Furthermore, we prove that there are only finitely many genus 2 curves (up to $\mathbb{C}$-isomorphism) defined over $\mathbb{Q}$, which have degree 2 and 3 elliptic subcovers also defined over $\mathbb{Q}$.

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