Abstract
Let \({\mathcal {O}}_K\) be a complete discrete valuation ring with algebraically closed residue field of positive characteristic and field of fractions \(K\). Let \(X_K\) be a torsor under an elliptic curve \(A_K\) over \(K\), \(X\) the proper minimal regular model of \(X_K\) over \(S:=\hbox {Spec}({\mathcal {O}}_K)\), and \(J\) the identity component of the Néron model of \(\mathrm{Pic}_{X_K/K}^{0}\). We study the canonical morphism \(q:\mathrm{Pic}^{0}_{X/S}\rightarrow J\) which extends the natural isomorphism on generic fibres. We show that \(q\) is pro-algebraic in nature with a construction that recalls Serre’s work on local class field theory. Furthermore, we interpret our results in relation to Shafarevich’s duality theory for torsors under abelian varieties.
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Acknowledgments
We thank M. Raynaud for suggesting this subject to us and for useful discussions. We thank the referee for the very detailed review of our paper. The preprint [28] was done when the second author was a post-doc in Essen, and he thanks H. Esnault for the hospitality. He also thanks D. Lorenzini, W. Zheng and C. Pépin for their remarks. He wants to thank especially M. Raynaud for allowing him to use his unpublished work [23], and for his generous help during the preparation of [28]. Both authors thank Progetto di Eccellenza Cariparo 2008–2009 “Differential methods in Arithmetics, Geometry and Algebra” for financial support.
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Bertapelle, A., Tong, J. On torsors under elliptic curves and Serre’s pro-algebraic structures. Math. Z. 277, 91–147 (2014). https://doi.org/10.1007/s00209-013-1247-5
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DOI: https://doi.org/10.1007/s00209-013-1247-5
Keywords
- Elliptic fibrations
- Models of curves
- Shafarevich pairing
- Abelian varieties
- Picard functor
- Pro-algebraic groups