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 give a short proof for the known fact the only possibilities of genera of such curves are 2, 3, 4 and 5; then we focus on the genus five 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. Our proof of the existence of hyperelliptic curves on general (1, 4)-polarised abelian surfaces is different from that in the recent paper [10]. 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.