Timeline for $p$-adic analogues of $\mathrm{SO}(3)$
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Jun 28, 2023 at 14:16 | history | edited | YCor | CC BY-SA 4.0 |
fixed typo, added tag
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| Aug 6, 2014 at 14:23 | comment | added | Keivan Karai | user52824: Thank you for the enlightening comment. I soon have to deal with several other cases (including different ${\mathbf Q}_p$-forms of ${\mathrm{SO}}_4$.) It would be great if you could please provide a reference that explains how to systematically find all these forms. | |
| Aug 5, 2014 at 15:10 | comment | added | user27920 | Forms of ${\rm{SO}}_3$ are the same as forms of ${\rm{PGL}}_2$, and those in turn correspond to isomorphism classes of quaternion algebras. By our knowledge of the Brauer group of local fields, there are exactly two such algebras over any local field. The corresponding groups are split (for the matrix algebra) and anisotropic (for the division algebra). | |
| Aug 5, 2014 at 14:04 | vote | accept | Keivan Karai | ||
| Aug 5, 2014 at 13:46 | answer | added | abx | timeline score: 7 | |
| Aug 5, 2014 at 13:44 | comment | added | YCor | No, since the dimension is odd, up to scaling you can suppose the determinant equal to $-1$, so the only remaining invariant is $\epsilon\in\{\pm 1\}$, and then for $\epsilon=1$ the form is isotropic (hence the group non-compact) and for $\epsilon=-1$ its anisotropic (and hence the group compact). | |
| Aug 5, 2014 at 13:27 | history | asked | Keivan Karai | CC BY-SA 3.0 |