Hyperelliptic curves on $(1,4)$ polarised abelian surfaces

TL;DR

This paper classifies hyperelliptic curves on general complex abelian surfaces, focusing on genus 5, and explores their geometric properties, automorphisms, and Jacobian decompositions, revealing unique invariance and structural features.

Contribution

It provides a complete classification of hyperelliptic curves of genera 2 to 5 on general abelian surfaces, especially characterizing genus 5 cases and their Jacobian structures.

Findings

Genus of hyperelliptic curves on general abelian surfaces is limited to 2, 3, 4, 5.

Unique genus 5 hyperelliptic curve exists up to translation on a (1,4)-polarized surface.

Every étale Klein covering of a hyperelliptic curve under certain conditions is hyperelliptic.

Abstract

We investigate the number and the geometry of smooth hyperelliptic curves on a general complex abelian surface. We show that the only possibilities of genera of such curves are $2,3,4$ and $5$. We focus on the genus 5 case. We prove that up to translation, there is a unique hyperelliptic curve in the linear system of a general $(1,4)$ polarised abelian surface. Moreover, the curve is invariant with respect to a subgroup of translations isomorphic to the Klein group. We give the decomposition of the Jacobian of such a curve into abelian subvarieties displaying Jacobians of quotient curves and Prym varieties. Motivated by the construction, we prove the statement: every \'etale Klein covering of a hyperelliptic curve is a hyperelliptic curve, provided that the group of $2$-torsion points defining the covering is non-isotropic with respect to the Weil pairing and every element of this group can be written as a difference of two Weierstrass points.