A Smooth Shioda-Tate Isomorphism for Elliptically Fibered 4-Manifolds

Abstract

This paper studies a subgroup of the mapping class group of an elliptically fibered $4$-manifold $π: M\to B$, inspired by number theory. The smooth Mordell-Weil group $\mathrm{MW}(π)$ is the group of isotopy classes of smooth sections of $π$. We construct a smooth analog of the Shioda-Tate isomorphism, identifying $\mathrm{MW(π)}$ with the quotient of $H^2(M;\mathbb{Z})$ by the trivial lattice of $π$. This shows that $\mathrm{MW}(π)$ can be computed explicitly from $H^2(M;\mathbb{Z})$. As an application, we show that the natural inclusion of the holomorphic Mordell-Weil group into the smooth one is an isomorphism if and only if $M$ is a rational elliptic surface. As another application, we show that $\mathrm{MW}(π)$ satisfies a strong version of Nielsen realization: any torsion element of $\mathrm{MW}(π)$ is represented by a holomorphic translation of $M$ of the same order.