On the $p^e$-torsion of elliptic curves and elliptic surfaces in characteristic $p$

We study the extension generated by the $x$-coordinates of the $p^e$-torsion points of an elliptic curve over a function field of characteristic $p$. If $S\to C$ is a non-isotrivial elliptic surface in characteristic $p$ with a $p^e$-torsion section, then for $p^e>11$ our results imply restrictions on the genus, the gonality, and the $p$-rank of the base curve $C$, whereas for $p^e\le 11$ such a surface can be constructed over any base curve $C$. We also describe explicitly all occurring $p^e$ in the cases where the surface $S$ is rational or $K3$ or the base curve $C$ is rational, elliptic or hyperelliptic.