Torsors under tori and N\'eron models

TL;DR

This paper investigates how torsors under tori over a local field extend to Néron models and how their evaluation maps relate to the special fiber, aiding the study of arithmetic properties of smooth varieties.

Contribution

It demonstrates that torsors under tori split by tamely ramified extensions extend to Néron models, linking their evaluation maps to the geometry of the special fiber.

Findings

Torsors under certain tori extend to Néron models.

Evaluation maps factor through the special fiber.

Geometry of the special fiber informs arithmetic properties.

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 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.