Elliptic Modules: Geometric Definition

Abstract

The definition in Chap. 3of elliptic modules as A−structures on the additive group \({\mathbb{G}}_{a,K}\) over a field K over A has a natural generalization in which the field K, that is, the scheme SpecK, is replaced by an arbitrary scheme S over A and \({\mathbb{G}}_{a,K}\) is replaced by an invertible (locally free rank one) sheaf \(\mathbb{G}\) over S (equivalently a line bundle over S). An elliptic module of rank r over S will then be defined as an A−structure on \(\mathbb{G}\) which becomes an elliptic module of rank r over K for any field K over S (thus SpecK→S). For our purposes it suffices to consider only affine schemes S and elliptic modules defined by means of a trivial line bundle \(\mathbb{G}\) alone.