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Néron models, Lie algebras, and reduction of curves of genus one. (English) Zbl 1060.14037
Invent. Math. 157, No. 3, 455-518 (2004); corrigendum ibid. 214, No. 1, 593-604 (2018).
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 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.

MSC:
14H25 Arithmetic ground fields for curves
14H40 Jacobians, Prym varieties
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