Torsion Sections of Elliptic Surfaces

In this article we discuss torsion sections of semistable elliptic surfaces defined over the complex field C. Recall that a semistable elliptic surface is a fibration π : X → C, where X is a smooth compact surface, C is a smooth curve, the general fiber of π is a smooth curve of genus one, and all the singular fibers of π are semistable, that is, all are of type Im in Kodaira’s notation (see [K]). In addition, we assume that the fibration π enjoys a section S0; this section defines a zero for a group law in each fiber, making the general fiber an elliptic curve over C. The Mordell-Weil group of X, denoted by MW(X), is the set of all sections of π, which is known to form a group under fiber-wise addition; the section S0 is the identity of MW(X). Note that any section S ∈ MW(X) meets one and only one component of each fiber. Now given any section S ∈ MW(X), one can ask the following two questions. First, which components of the singular fibers of X does S meet, and second, exactly where in these components does S meet them? When S is a torsion section, the first question was addressed by Miranda in [M]. To describe those results we require some notation. Recall that a singular semistable fiber of type Im is a cycle of m P ’s. Suppose that our elliptic fibration π : X → C has s such singular fibers F1, . . . , Fs, with Fj of type Imj . Choose an “orientation” of each fiber Fj and write the mj components of Fj as