Timeline for abelianization of adelic points of an algebraic group
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 24, 2013 at 16:15 | comment | added | Venkataramana | For a reference, there is a paper by Raghunathan and Gopal Prasad on $SL_1(D)$. | |
| Feb 24, 2013 at 16:14 | comment | added | Venkataramana | Over a local field, take a quaternionic division algebra $D$. Then $D$ contains every quadratic subfield of $K$ (this is known; Platonov-Rapinchuk also state this). In particular, $D$ contains the unique unramified quadratic extension $E/K$. Moreover, the nontirvial Galois element of $E/K$ also lies in $D$. If $\pi$ is the uniformising parameter of $K$, it is also a uniformising parameter for $E$ and one can see that $SL_1(D)$ modulo the congruence subgroup of level one with respect to $\pi$ is the group of norm one elements of $E/K$ plus the Galois automorphism; this is solvable. | |
| Feb 24, 2013 at 15:25 | comment | added | YCor | @Aakumadula: thanks for the correction. I should have assumed the group has no $K$-anisotropic factor; I edit accordingly. I don't know much about the anisotropic case so I'm not able to give an explicit example of a division algebra over a number field for which $SL_1(D)$ is not a perfect group but I'm ready to believe it indeed exists. | |
| Feb 24, 2013 at 15:21 | history | edited | YCor | CC BY-SA 3.0 |
added 39 characters in body
|
| Feb 24, 2013 at 14:06 | comment | added | Venkataramana | This is not quite correct. You can take $G=SL_1(D)$ over $K_v$ where $D$ is a central division algebra over $K_v$. In general, it is not its own commutator; however, the algebraic group is semi-simple and simply connected. | |
| Jun 29, 2012 at 13:07 | vote | accept | Judith Ludwig | ||
| Jun 28, 2012 at 19:53 | history | answered | YCor | CC BY-SA 3.0 |