Genus 2 curves that admit a degree 5 map to an elliptic curve

K. Magaard; T. Shaska; H. Voelklein

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arXiv · 1209.0443

Genus 2 curves that admit a degree 5 map to an elliptic curve

K. Magaard·T. Shaska·H. Voelklein

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Abstract

We continue our study of genus 2 curves $C$ that admit a cover $ C \to E$ to a genus 1 curve $E$ of prime degree $n$. These curves $C$ form an irreducible 2-dimensional subvariety $Ł_n$ of the moduli space $\M_2$ of genus 2 curves. Here we study the case $n=5$. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general $n$. We compute a normal form for the curves in the locus $Ł_5$ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of $Ł_5$ as subvarieties of $\M_2$ and classify all curves in these loci which have extra automorphisms.

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BibTeX RIS

K. Magaard, T. Shaska, H. Voelklein. 2012-09-03. Genus 2 curves that admit a degree 5 map to an elliptic curve. https://arxiv.org/abs/1209.0443

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