Mordell--Weil groups of quasi-elliptic surfaces in characteristic 3

In characteristic $p=3$, we establish a concrete relationship between the torsion rank of the elementary abelian group of global sections of quasi-elliptic surfaces over the projective line ${\mathbf{P}}^1$ and the $p$-rank of certain hyperelliptic curves, thereby explaining a mysterious relationship between these two invariants previously observed by I. Dolgachev. As an application, we determine the torsion ranks of the Mordell–Weil groups of a class of quasi-elliptic surfaces.