I have found that a lot of research has been done in rationality problem for algebraic tori. (For example, https://arxiv.org/abs/1210.4525). So I got to wonder what historical context or elementary motivation the problem has.
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$\begingroup$ One possible reason is that the rationality problem for varieties is interesting but typically pretty hard, and tori are varieties that we understand very well. But I imagine there are reasons coming from number theory as well. $\endgroup$– R. van Dobben de BruynCommented Mar 13, 2021 at 2:19
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$\begingroup$ Rationality of tori is a key ingredient in the approach to "local-global problems" via "patching" as in the work of Colliot-Th'el`ene -- Harbater -- Hartmann -- Krashen -- Parimala -- Suresh. It also comes up in work on weak approximation and strong approximation for linear algebraic groups and for homogeneous varieties under the action of a linear algebraic group. I am sure that Mikhail Borovoi could write more if he sees this post. $\endgroup$– Jason StarrCommented Mar 13, 2021 at 11:21
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Algebraic tori – thirty years after gives some historical context:
The rationality problem goes back to the study of Pythagorean triples : Describe the set of solutions of a given system of polynomial equations by rational functions in a certain number of parameters. To establish rationality, one usually has to exhibit some explicit parameterization such as that obtained by stereographic projection in the case of Pythagoreas triples.
In the converse "nonrationality" problem one wants to establish the non-existence of such a parameterization: here one seeks a birational invariant allowing one to detect nonrationality by comparing its value for the object under consideration with some “standard” one known to be zero; if the computation gives a nonzero value, we are done. Evidently, to be useful, such an invariant must be (relatively) easily computable.
Algebraic tori are among the simplest algebraic groups, so much of the search for birotational invariants has focused on that topic. Pioneering results by Voskresenskii are reviewed in this paper.
answered Mar 13, 2021 at 9:10