Timeline for For what fields is $GL_n(k)$ a rational variety?
Current License: CC BY-SA 3.0
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| Jun 25, 2013 at 3:02 | review | First posts | |||
| Jun 25, 2013 at 23:19 | |||||
| Jun 16, 2013 at 9:22 | comment | added | Jérémy Blanc | The question seems interesting but the title should maybe be changed. It is obvious that $GL_n$ is rational over any field, as Abhinav Kumar pointed out, but there are linear groups which are not rational (although they are geometrically rational), so the question is probably about these ones. And it is true, as ayanta pointed out, that an algebraic group is not only its set of rational points. | |
| Jun 16, 2013 at 6:01 | comment | added | user30180 |
The set of rational points has no "algebro-geometric structure". For example, ${\rm{GL}}_n(\mathbf{Q}_p)$ is a group, or even $p$-adic analytic manifold, but it is not a variety. The $\mathbf{Q}_p$-variety ${\rm{GL}}_n$ (denoted $({\rm{GL}}_n)_{\mathbf{Q}_p}$ if you wish) is what is rational, not its set of $\mathbf{Q}_p$-points. Please be clear with the notation. See the paper "Rationality problem for semisimple group varieties" by Chernousov and Platonov for the non-split case, especially the Main Theorem in section 1 (and type-A counterexamples in rank $> 2$, due to Merkurjev and Rost).
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| Jun 16, 2013 at 4:12 | comment | added | Geordie Williamson | More generally, any split reductive algebraic group is rational, as follows from the existence of the big cell (an open dense set isomorphic to the product of a Borel subgroup (isomorphic to a variety to a product of affine lines and affine lines minus zero) and the unipotent radical of the opposite Borel (isomorphic to affine space). | |
| Jun 16, 2013 at 4:08 | comment | added | Keerthi Madapusi | What is your definition of rational? | |
| Jun 16, 2013 at 3:05 | comment | added | Abhinav Kumar | $GL_n$ is certainly a rational variety (over any field), since it's birational to $M_n$ which is affine $n^2$ space. | |
| Jun 16, 2013 at 1:29 | history | asked | Anna | CC BY-SA 3.0 |