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

TL;DR

This work resolves a Prym–Torelli-type question for unramified $Z_3 imesZ_3$-coverings of genus-2 curves by proving that the Prym variety determines the covering on both isotropic and non-isotropic components. The authors achieve this via a polarization-driven reconstruction: they recover the $Z_3 imesZ_3$-action from the Prym polarization, decompose the Prym into four small Pryms, extract a rich set of elliptic curves, and recover nine lifts of the hyperelliptic involution to reconstruct genus-4 Jacobians and thus the original covering. Beyond the genus-2 case, they analyze Prym maps for arbitrary abelian G-coverings, establishing generic finiteness except for small cyclic groups, and delineating non-emptiness conditions tied to the rank of $G$. Overall, the paper extends Prym–Torelli results to a new abelian-quotient setting and clarifies when Prym maps are finite versus generically finite in the abelian regime, offering tools for reconstructing coverings from Prym data in genus two.

Abstract

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

Theorems & Definitions (41)

• Theorem 1.1: Theorem
• Theorem 1.2: Theorem
• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Definition 2.4
• Lemma 2.5