The smooth Mordell-Weil group and mapping class groups of elliptic surfaces

Abstract

This is a paper in smooth $4$-manifold topology, inspired by the N\'{e}ron-Lang Theorem in number theory. More precisely, we prove that a smooth version $\MW(\pi)$ of Mordell-Weil group of an elliptic fibration $\pi:M\to\Pb^1$ is finitely generated. We compute $\MW(\pi_d)$ explicitly for elliptic fibrations $\pi_d:M_d\to\Pb^1$, where $M_d$ is a simply-connected complex surfaces $M_d$ of arithmetic genus $d\geq 1$ and all fibers of $\pi_d$ are nodal. We prove in this case that the fibered structure is unique up topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of $M_d$ with $d\geq 3$ is topologically isotopic to a diffeomorphism taking fibers to fibers.