Rational curves on varieties over finite fields

I will present recent work of Bogomolov and Tschinkel, with a simplification due to myself. Let $k$ be an algebraic closure of a finite field. If $X$ is a curve over $k$ of positive genus embedded in its Jacobian $J$ using a basepoint, the union of the images of $X$ under endomorphisms of $J$ contains $J(k)$. This can be used to construct a K3 surface $S$ such that every $k$-point of $S$ lies on a rational curve in $S$, though the (geometric) generic point of $S$ does not lie on a rational curve. Such behavior is impossible over ${\mathbb C}$.