Curves over finite fields

Consider a curve $C$ in abelian variety $A$ defined over over an algebraic closure of finite field (with fixed zero). Then 1) there is a set of primes $S$ of density one (including also any given finite set of primes) such the projection of $C$ on $S$-primary subgroup of $A$ is surjetive (recall that $A$ is a torsion group which is product of primary division groups corresponding to different primes — we consider a product of all such groups for primes in $S$).
In particular for any point $x\in A$ there is a integer $N$ and point $c\in C$ such that $Nc = x$. I will also discuss some related geometric results (rational connectedness of Kummer varieties and many other over algebraic closure of a finite field).