Torsion Sections of Semistable Elliptic Surfaces

In this article we would like to address the following situation. Let f : X → C be an elliptic surface with section S0. We further assume that X is relatively minimal and smooth, and that all singular fibers are semistable, that is, are cycles of P’s (type “Im” in Kodaira’s notation, see [K]). Given a section S of f , it will meet one and only one component in every singular fiber. The fundamental aim of this paper is to try to say something specific about which components are being hit, in the case that S is a torsion section (i.e., of finite order in the Mordell-Weil group of sections of f). Most work on the Mordell-Weil group for elliptic surfaces focuses on properties of the group itself, (for example its rank, etc.) and not on properties of the elements of the group. See for example [Shd] and [C-Z] for general properties, and papers such as [Sch] and [Stl] for computations in specific cases. Although the methods used in this article are elementary, they are suprisingly powerful, enabling us to obtain rather detailed information about the possibilities for component intersection for a torsion section. To be specific, suppose that the j singular fiber is a cycle of length mj, and that the components are numbered cyclically around the cycle from 0 to mj − 1. Suppose that in this j fiber, a section S of order exactly n hits the k j component. The main restrictions on these “component” numbers kj are the following: