# Torsors under tori and Néron models

• [1] Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL, UK
• Volume: 23, Issue: 2, page 309-321
• ISSN: 1246-7405

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## Abstract

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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 fibre to study the arithmetic of $X$.

## How to cite

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Bright, Martin. "Torsors under tori and Néron models." Journal de Théorie des Nombres de Bordeaux 23.2 (2011): 309-321. <http://eudml.org/doc/219777>.

@article{Bright2011,
abstract = {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 $\mathcal\{X\}/R$ is a model of $X$ then, using a result of Brahm, we show that $X$-torsors under $G$ extend to $\mathcal\{X\}$-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 fibre to study the arithmetic of $X$.},
affiliation = {Mathematics Institute Zeeman Building University of Warwick Coventry CV4 7AL, UK},
author = {Bright, Martin},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Torsors; Néron models; torsor; Néron model},
language = {eng},
month = {6},
number = {2},
pages = {309-321},
publisher = {Société Arithmétique de Bordeaux},
title = {Torsors under tori and Néron models},
url = {http://eudml.org/doc/219777},
volume = {23},
year = {2011},
}

TY - JOUR
AU - Bright, Martin
TI - Torsors under tori and Néron models
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2011/6//
PB - Société Arithmétique de Bordeaux
VL - 23
IS - 2
SP - 309
EP - 321
AB - 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 $\mathcal{X}/R$ is a model of $X$ then, using a result of Brahm, we show that $X$-torsors under $G$ extend to $\mathcal{X}$-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 fibre to study the arithmetic of $X$.
LA - eng
KW - Torsors; Néron models; torsor; Néron model
UR - http://eudml.org/doc/219777
ER -

## References

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