Néron models, Lie algebras, and reduction of curves of genus one (Q1882581)

Let \(\mathcal O_K\) be the ring of integers of a discrete valuation field \(K\) whose residue field \(k\) is of characteristic \(p \geq 0\). Let \(X_K\) be a smooth connected projective curve of genus one over \(K\), and let \(E_K\) be the Jacobian of \(X_K\). Let \(X/S\) and \(E/S\) with \(S= \text{Spec} (\mathcal O_K)\) be the minimal regular models of \(X_K\) and \(E_K\), respectively. In this paper the authors investigate possible relationships between the special fibers \(X_k\) and \(E_k\) by studying the geometry of the Picard functor \(\text{Pic}_{X/S}\) when \(X/S\) is not necessarily cohomologically flat. As an application they prove in full generality a theorem of \textit{W.~Gordon} [Compos. Math. 38, 163--199 (1979; Zbl 0425.14003)] on the equivalence between the Artin-Tate conjecture and the Birch-Swinnerton-Dyer conjecture.