Let $Y$ be an abelian surface. Is it true that for every general point $P \in Y$, there exists an elliptic curve passing through $P$?



Elliptic curves on abelian surface
In general, if an abelian variety $A$ contains an abelian subvariety $B\subseteq A$, then $A$ contains another abelian subvariety $B'\subseteq A$ such that $A$ is isogenous to $B\times B'$. This is [Poincaré's reducibility theorem](http://books.google.com/books?id=MOW2gEP7HIkC&lpg=PP1&dq=birkenhake%2520lange&pg=PA125#v=onepage&q=poincare%2520reducibility&f=false). (See also Poincaré's complete reducibility theorem, same book, next page). 


An abelian variety is called *simple* if it does not contain any nontrivial abelian subvarieties. Simon's argument shows that there exist simple complex tori of dimension 2. 
One could also count moduli and conclude that not every abelian surface (or abelian variety of arbitrary dimension $>1$) can be isogenous to a product. 


