The Smooth Narrow Mordell-Weil Group of Elliptically Fibered 4-Manifolds

Abstract

For an elliptically fibered 4-manifold $\pi\colon M\to B$, we introduce a subgroup of the smooth mapping class group of $M$ with explicit geometric representatives: the smooth narrow Mordell-Weil group $MW_0(\pi)$. Leveraging Kodaira's classification of singular fibers, we construct a natural isomorphism between $MW_0(\pi)$ and the orthogonal complement of the trivial lattice in $H^2(M)$. This identification parallels its holomorphic counterpart. This paper generalizes work of Farb and Looijenga, who defined the smooth Mordell-Weil group and computed it for fibrations over a sphere with only nodal fibers. We give an explicit formula for the rank of $MW_0(\pi)$ depending only on the genus of the base curve and the types of singular fibers of $\pi$. As a corollary, we show that rational elliptic surfaces are the only elliptic surfaces with holomorphic and smooth Mordell-Weil groups of the same rank.