  We study smooth curves on abelian surfaces, especially for genus 4, when the
complementary subvariety in the Jacobian is also a surface. We show that up to
translation there is exactly one genus 4 hyperelliptic curve on a general (1,
3)-polarised abelian surface. We investigate these curves and show that their
Jacobians contain a surface and its dual as complementary abelian subvarieties.
