On the Noether Problem for torsion subgroups of tori

Federico Scavia

Vol. 306 (2020), No. 2, 699–719

DOI: 10.2140/pjm.2020.306.699

Abstract

We consider the Noether Problem for stable and retract rationality for the sequence of $d$-torsion subgroups $T\left[d\right]$ of a torus $T$, $d\ge 1$. We show that the answer to these questions only depends on $d\left(mode\left(T\right)\right)$, where $e\left(T\right)$ 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 $T\left[d\right]$. We also give an application to the Grothendieck ring of stacks.

Keywords

Noether problem, algebraic tori, retract rationality

Mathematical Subject Classification 2010

Primary: 14L15, 14M20

Milestones

Received: 1 May 2019

Revised: 25 December 2019

Accepted: 2 January 2020

Published: 13 July 2020