  This is a paper in smooth $4$-manifold topology, inspired by the Mordell-Weil
Theorem in number theory. More precisely, we prove a smooth version of the
Mordell-Weil Theorem and apply it to the `unipotent radical' case of a
Thurston-type classification of mapping classes of simply-connected
$4$-manifolds $M_d$ that admit the structure of an elliptic complex surface of
arithmetic genus $d\geq 1$. Applications include Nielsen realization theorems
for $M_d$.
  By combining this with known results, we obtain the following remarkable
consequence: if the singular fibers of such an elliptic fibration are of the
simplest (i.e.\ nodal) type, then the fibered structure is unique up
topological isotopy. In particular, any diffeomorphism of $M_d,d\geq 3$ is
topologically isotopic to a diffeomorphism taking fibers to fibers.
