| I will report on the results of an ongoing project which we began some years ago with Yuri Tschinkel and continue with Hang Fu and Jin Qian. We say that a smooth projective curve C dominates C' if there is nonramified covering of C which has a surjection onto C'. Thanks to Bely theorem we can show that any curve C' defined over $\bar Q$ is dominated by one of the curves $C_n, y^n-1= x^2$. Over $\bar F_p$ any curve in fact is dominated by $C_6$ which is in way also a minimal possible curve with such a property. Conjecturally the same holds over $\bar Q$ but at the moment we can prove only partial results in this direction. There are not many methods to establish dominance for a particular pair of curves and the one we use is based on the study of torsion points and finite unramified covers of elliptic curves. |