You may have a look to:

[Ernst Kani, *Elliptic curves on Abelian surfaces*][1]

(the credit for this reference goes to Dan Petersen who already suggested it in a comment to [this][2] question) 


  [1]: http://www.mast.queensu.ca/~kani/papers/hum-msm.pdf
  [2]: https://mathoverflow.net/questions/21439/which-curves-can-be-found-on-abelian-varieties