Torsors under tori and Néron models

Details Torsors under tori and Néron models Torsors under tori and Néron models

2009-11-25

arxiv.org/abs/0911.4834v2

Journal de Th\'eorie des Nombres de Bordeaux 23 (2011), 309-321

Mathematics

Algebraic Geometry

10 pages

Scientific paper

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 XX/R is a model of X then, using a result of Brahm, we show that X-torsors under G extend to XX-torsors under a N\'eron 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 fibre to study the arithmetic of X.

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