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 Lang–Néron theorem in number theory. More precisely, we prove that a smooth version MW(π) of the Mordell–Weil group of an elliptic fibration π: M → P1 is finitely generated. We compute MW(πd) explicitly for elliptic fibrations πd: Md → P1, where Md is a simply connected complex surface of arithmetic genus d ≽ 1 and all fibers of πd are nodal. We prove in this case that the fibered structure is unique up to topological isotopy. By combining this with a result of Donaldson, we obtain the following remarkable consequence: any diffeomorphism of Md with d ≽ 3 is topologically isotopic to a diffeomorphism taking fibers to fibers.