Elliptic Surfaces

Abstract

Let E be an elliptic surface having a global section over its base curve X. We assume throughout that the functional invariant J is non-constant and that E has no exceptional curves of the first kind in the fibers. We shall frequently refer to such an elliptic surface as a basic elliptic surface. The generic fiber E^gen is naturally an elliptic curve over the function field K(X) of the base curve with the section providing a K(X)-rational point to serve as zero in the group. We shall be interested in the group of K(X)-rational points on E^gen, denoted E^gen (K(X)), which corresponds to the group of sections of E over X. This is a finitely generated abelian group whose rank we denote by r.