Timeline for Classification of Tori of GL2, up to conjugation
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 18, 2013 at 21:01 | comment | added | Jérémy Blanc | @ Shripad, yes I see that torus can be complicated in general over Q (more than over R); I just meant that in GL2 we can have an explicit description (given above in the answer of Will). Thx anyway for your answers. | |
| Feb 18, 2013 at 6:59 | comment | added | Shripad | Thanks Will, your comment elaborates my answer very well. Jeremy, by complicated for $\mathbb{Q}$ I meant only in comparison with the case over $\mathbb{R}$. | |
| Feb 17, 2013 at 0:46 | comment | added | Jérémy Blanc | Ok, thx. It seems thus that 2-dim tori are either diagonalisable or of the form [a,muc,c,a] with a,c \in k^{}. And 1-dimensional tori can be found in a similar way. | |
| Feb 16, 2013 at 15:44 | comment | added | Will Sawin | You can do this via the explicit description of all quadratic characters of $GL_2(\mathbb Q)$ coming from Kummer theory (or class field theory). In particular, every quadratic character trivializes over $\mathbb Q(\sqrt{D})$ for some $D$, and the corresponding torus is the group of determinant $1$ matrices that preserve the quadratic form $x^2-Dy^2$, as was first pointed out by Robert Bryant. | |
| Feb 16, 2013 at 11:00 | comment | added | Jérémy Blanc | Thanks for the comment. I would like to use the fact that I am restricted to GL2 to really have all tori. For example, over $\mathbb{Q}$ the results should not be so complicated. | |
| Feb 16, 2013 at 7:23 | history | answered | Shripad | CC BY-SA 3.0 |