Klein coverings of genus 2 curves

Paweł Borówka; Angela Ortega

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arXiv · 1904.05962

Klein coverings of genus 2 curves

Paweł Borówka·Angela Ortega

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Abstract

We investigate the geometry of étale $4:1$ coverings of smooth complex genus 2 curves with the monodromy group isomorphic to the Klein four-group. There are two cases, isotropic and non-isotropic depending on the values of the Weil pairing restricted to the group defining the covering. We recall from our previous work \cite{bo} the results concerning the non-isotropic case and fully describe the isotropic case. We show that the necessary information to construct the Klein coverings is encoded in the 6 points on $\mathbb{P}^1$ defining the genus 2 curve. The main result of the paper is the fact that, in both cases the Prym map associated to these coverings is injective. Additionally, we provide a concrete description of the closure of the image of the Prym map inside the corresponding moduli space of polarised abelian varieties.

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BibTeX RIS

Paweł Borówka, Angela Ortega. 2019-11-12. Klein coverings of genus 2 curves. https://arxiv.org/abs/1904.05962

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