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Let $G$ be a finite group, and let $V$ be a faithful representation of $G$. The Noether problem asks whether $V/G$ is rational (stably rational, retract rational) or not.

To construct counterexamples to the Noether problem is usually very difficult, and it is done by means of birational invariants (e.g. unramified cohomology). I was told that recently new invariants are being developed, using $\mathbb{A}^1$-homotopy theory.

I would like to learn about these developments, but a search online did not yield anything. Note that I know very little about motivic homotopy theory, but this is not a problem, because I intend to devote the time necessary to learn it (keeping the Noether problem in mind as my personal goal).

Do you know where these invariants are defined, and where the theory is built?

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    $\begingroup$ A possible reference is the paper "Smooth varieties up to A1-homotopy and algebraic h-cobordisms" by Asok and Morel. You could also look up the papers that cite this paper in mathscinet. $\endgroup$
    – naf
    Commented Jul 20, 2018 at 5:05
  • $\begingroup$ @ulrich the OP may not have access to MathSciNet... Google Scholar may do an acceptable job if so. $\endgroup$
    – David Roberts ♦
    Commented Jul 20, 2018 at 14:08

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