#### Vol. 306, No. 2, 2020

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On the Noether Problem for torsion subgroups of tori

### Federico Scavia

Vol. 306 (2020), No. 2, 699–719
##### 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 $d\phantom{\rule{0.3em}{0ex}}\left(mod\phantom{\rule{0.3em}{0ex}}e\left(T\right)\right)$, 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.

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