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

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.