Pryms of ${\mathbb{Z}}_3\times {\mathbb{Z}}_3$ coverings of genus 2 curves

We study unramified Galois ${\mathbb{Z}}_3\times {\mathbb{Z}}_3$ coverings of genus 2 curves and the corresponding Prym varieties and Prym maps. In particular, we prove that any such covering can be reconstructed from its Prym variety, that is, the Prym-Torelli theorem holds for these coverings. We also investigate the Prym map of unramified $G$-coverings of genus 2 curves for an arbitrary abelian group $G$. We show that the generic fiber of the Prym map is finite unless $G$ is cyclic of order less than 6