# 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.

## Key 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.