Which curves can be found on Abelian varieties? We know that each genus 2 curve is embedded into its degree 1 Jacobian. 
Under which conditions on $C$, $A$, $g$ and $n$ is it possible for a genus $g$ smooth curve $C$ to be embedded in an Abelian variety $A$ of dimension $n$?
What can be said in the case $n=2$? 
And what if $A$ is a Jacobian (and e.g. $n=2$)?
 A: Gian Pietro Pirola in  [Curves on generic Kummer varieties, 
 Duke Math. J. Volume 59, Number 3 (1989), 701-708] proves a rigidity theorem for curves of  genus $g\le q-2$ in the Kummer variety of a $q$-dimensional abelian variety. As a consequence, he shows that a generic abelian variety of dimension $q\ge 3$ does not contain hyperelliptic curves of any genus.
A: In his paper "Abelian surfaces with (1,2)-polarization" Barth proves that a smooth genus $3$ curve can be embedded into an Abelian surface if and only if it admits an elliptic involution, i.e. a map of degree $2$  onto an elliptic curve.  
A: For any $g >0$ there exists an abelian surface $A$ containing a smooth curve $C$ of genus $g$; the surface can be assumed to be simple if $g > 1$: 
For $g=1$, one can just consider $C \times C$. For $g>1$, consider the generic abelian surface with a polarisation of type $(1, g-1)$. By the adjunction formula, a smooth curve in the linear system corresponding to the polarisation will have genus $g$. (Such abelian surfaces can be constructed by considering suitable quotients of principally polarised ones ($n=1$))
One gets abelian varieties of any dimension $>1$ by taking products.
However, this does not produce a fixed abelian surface containing curves of all genera $g>1$ but it seems likely that such surfaces should exist. It would be interesting to know whether this is possible for the generic principally polarised surface, or more generally ppav of any dimension $n$.
A: If $C$ has a map onto a smooth curve $B$, there is a surjective map $J(C)\to J(B)$ hence $C$ maps to $J(B)$ and it is probably often an embedding.
