abstract: I will discuss a number of results on the structure of the image of a curve defined over $\bar Fp$ in its jacobian. Te latter is viewed as an infinite torsion group. One of the applications of these results is "rational connectedness" of all Kummer $K3$ surfaces over $\bar Fp$. Namely for any finite set of points in Kummer a $K3$ surfaces there is a chain rational curves connecting thes points. An "approximation" version of this result holds also for $\bar Q$-points for Kummer $K3$ surfaces defined over $\bar Q$.
It is a joint works with Yu.Tschinkel.