We study smooth curves on abelian surfaces, especially for genus 4 4 , when the complementary subvariety in the Jacobian is also a surface. We show that up to translation there is exactly one genus 4 4 hyperelliptic curve on a general ( 1 , 3 ) (1,3) -polarised abelian surface. We investigate these curves and show that their Jacobians contain a surface and its dual as complementary abelian subvarieties.
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Borówka et al. (2017) studied this question.