# Klein coverings of genus 2 curves

## TL;DR

This paper studies special 4:1 coverings of genus 2 curves with Klein group symmetry, describing their geometry, classification, and the injectivity of the associated Prym map, linking coverings to moduli spaces.

## Contribution

It fully characterizes isotropic Klein coverings of genus 2 curves and proves the Prym map's injectivity for these coverings, extending previous results.

## Key findings

- The Prym map for Klein coverings is injective.
- The geometry of Klein coverings is determined by 6 points on P^1.
- The closure of the Prym map's image is explicitly described.

## Abstract

We investigate the geometry of \'etale $4:1$ 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.