Néron models, Lie algebras, and reduction of curves of genus one

Abstract

Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically connected projective curve of genus 1 over K . Denote by EK the Jacobian of X K . Let X/S and E/S be the minimal regular models of X K and EK , respectively. In this article, we investigate the possible relationships between the special fibers Xk and Ek . In doing so, we are led to study the geometry of the Picard functor Pic X/S when X/S is not necessarily cohomologically flat. As an application of this study, we are able to prove in full generality a theorem of Gordon on the equivalence between the Artin-Tate and Birch-SwinnertonDyer conjectures. Recall that when k is algebraically closed, the special fibers of elliptic curves are classified according to their Kodaira type, which is denoted by a symbol T ∈ {In, I∗n, n ∈ Z≥0, II, II∗, III, III∗, IV, IV∗}. Given a type T and a positive integer m, we denote by mT the new type obtained from T by multiplying all the multiplicities of T by m. When k is algebraically closed, the relationships between the type of a curve of genus 1 and the type of its Jacobian can be summarized as follows.