# On the Noether Problem for torsion subgroups of tori

## TL;DR

This paper investigates the Noether Problem for torsion subgroups of algebraic tori, revealing that the problem's outcome depends only on the torsion order modulo the period of the generic torsor, with specific results for norm one tori.

## Contribution

It establishes a dependence of the Noether Problem's answer on the torsion order modulo the period and characterizes solutions for norm one tori, also connecting to the Grothendieck ring of stacks.

## Key findings

- Answer depends only on d mod e(T)
- Identifies all d with positive solutions for norm one tori
- Provides applications to the Grothendieck ring of stacks

## Abstract

We consider the Noether Problem for stable and retract rationality for the sequence of $d$-torsion subgroups $T[d]$ of a torus $T$, $d\geq 1$. We show that the answer to these questions only depends on $d\pmod{e(T)}$, where $e(T)$ is the period of the generic $T$-torsor. When $T$ is the norm one torus associated to a finite Galois extension, we find all $d$ such that the Noether Problem for retract rationality has a positive solution for $d$. We also give an application to the Grothendieck ring of stacks.