# Local-global principles for norm one tori over semi-global fields

## TL;DR

This paper establishes local-global principles for norm one tori over semi-global fields, linking Galois extensions and valuation theory under specific conditions on the residue field and curve geometry.

## Contribution

It proves a local-global principle for norms from Galois extensions over semi-global fields, extending previous results to new geometric and field conditions.

## Key findings

- Local-global principles hold under certain residue field conditions
- Norms from Galois extensions can be characterized globally via local data
- Results apply to fields with algebraically closed or finite residue fields

## Abstract

Let K be a complete discretely valued field with residue field k and F be a function field of a curve over K. Let L/F be a Galois extension of degree n. If n is coprime to char(k), then under some assumptions on k(e.g. k is algebraically closed or a finite field) and on the geometry of the curve, we show that there is a local-global principle with respect to discrete valuations for the norms from the extension L/F.