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Torsors under tori and Néron models

Martin Bright — 2011

Journal de Théorie des Nombres de Bordeaux

Let R be a Henselian discrete valuation ring with field of fractions K . If X is a smooth variety over K and G a torus over K , then we consider X -torsors under G . If 𝒳 / R is a model of X then, using a result of Brahm, we show that X -torsors under G extend to 𝒳 -torsors under a Néron model of G if G is split by a tamely ramified extension of K . It follows that the evaluation map associated to such a torsor factors through reduction to the special fibre. In this way we can use the geometry of the special...

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