ar X iv : a lg - g eo m / 9 21 00 09 v 2 1 6 A pr 1 99 3 Elliptic Three - folds I : Ogg - Shafarevich Theory

Let S be a variety over an algebraically closed field k of characteristic zero, with function field K, and let A be an elliptic curve over K. The Weil-Châtelet group of A, WC(A), is the set of principal homogeneous spaces (torsors) of A over K, i.e. isomorphism classes of curves of genus 1 over K which have A as their jacobian. This classifies, up to birational equivalence, elliptic fibrations over the variety S with the same jacobian. OggShafarevich theory was developed independently in [22] and [25] to calculate this group in the case that S is a curve. In this paper, we develop certain aspects of this theory in higher dimensions. In §1, we define the Tate-Shafarevich group, which is the subset of WC(A), XS(A), defined as follows. Let E ∈ WC(A) and let f : X → S be an elliptic fibration with generic fibre equal to E. Then E ∈ XS(A) if for all s ∈ S there exists an étale neighborhood U → S of s such that X ×S U → U has a rational section. Thus, in particular, f does not have multiple fibres in codimension one. We show how to calculate XS(A) when we have a model f : X → S of some torsor E of A where X and S are smooth and has fibres only of dimension one. We relate XS(A) to the Brauer group of X and S. In §2 we consider several different questions. First, we discuss when one can find a suitable model as above to calculate the Tate-Shafarevich group. Secondly, we wish to refine our interpretation of the Tate-Shafarevich group by giving conditions for when an element E ∈ XS(A) has a model f : X → S which has no isolated multiple fibres whatsoever. For the first question, we can construct a model for A over a base which might have to be a blowup of the original base, following [17], to obtain a so-called Miranda model which satisfies the required hypotheses to calculate the Tate-Shafarevich group. For the second question, it turns out that as long as A has a Miranda model over S, any element