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arXiv · 1209.0434

Families of genus two curves with many elliptic subcovers

Tony Shaska
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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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Tony Shaska. 2012-09-03. Families of genus two curves with many elliptic subcovers. https://arxiv.org/abs/1209.0434

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