Timeline for Compact form of symplectic groups defined over the rationals
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 24, 2015 at 11:11 | comment | added | Sebastian Schoennenbeck | Thank you for your comments, they were very helpful. I came across the construction via the quaternions earlier but was somehow thrown off the tracks by a a confusing remark. | |
| Jun 24, 2015 at 11:08 | vote | accept | Sebastian Schoennenbeck | ||
| Jun 24, 2015 at 4:18 | comment | added | Mikhail Borovoi | @YCor: Moreover, any $\mathbb{Q}$-form of ${\rm Sp}_{2m}$ splits at almost all primes $p$ (because it is an inner form of a split $\mathbb{Q}$-group). | |
| Jun 24, 2015 at 4:10 | comment | added | Mikhail Borovoi | I guess that the standard quaternion algebra $H=\mathbb{Q}(i,j)$ with $i^2=-1,\ j^2=-1,\ ji=-ij$ ramifies exactly at $\infty$ and 2. It follows that the unitary group $G=SU(H^n,F)$ of the hermitian form $F(x)=x_1 \bar{x}_1+\dots+x_n\bar{x}_n$ from YCor's first comment splits at every prime $p$ except $\infty$ and maybe at $p=2$. It does not split at 2 because it cannot be nonsplit at one place only by the reciprocity law (I mean the Hasse-Brauer-Noether theorem). | |
| Jun 23, 2015 at 21:38 | comment | added | YCor | PS: I guess that a semisimple group over $\mathbf{Q}$ will be automatically be split over $\mathbf{Q}_p$ for $p$ ranging over a positive density set of primes. | |
| Jun 23, 2015 at 14:38 | answer | added | grghxy | timeline score: 7 | |
| Jun 23, 2015 at 14:02 | comment | added | YCor | All simply connected semisimple real algebraic groups are definable over the rationals. Here the simplest is just the stabilizer of the standard positive definite hermitian form on the $H^n$ where $H$ are the usual quaternions (with the standard basis). | |
| Jun 23, 2015 at 13:40 | history | asked | Sebastian Schoennenbeck | CC BY-SA 3.0 |