Timeline for Number of points on a linear algebraic group over a finite field
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 16, 2023 at 21:08 | history | edited | LSpice | CC BY-SA 4.0 |
A little more clarification
|
| Apr 16, 2023 at 21:03 | comment | added | LSpice | Re, of course linear algebraic groups possess maximal tori by dimension counting. I should have said that they possess geometrically maximal tori, i.e., maximal tori that remain maximal after base change to an algebraic closure, by Grothendieck's theorem. | |
| Apr 16, 2023 at 20:52 | comment | added | LSpice | @DanielLoughran, re, thanks for the suggestion! I have done so. | |
| Apr 16, 2023 at 20:51 | history | edited | LSpice | CC BY-SA 4.0 |
Responding to @DanielLoughran's comment https://mathoverflow.net/questions/444855/number-of-points-on-a-linear-algebraic-group-over-a-finite-field/444877#comment1148974_444877
|
| Apr 16, 2023 at 20:26 | comment | added | Daniel Loughran | For completeness it would also be nice to explain how to prove the lower bound for algebraic tori, which seems to be the key case. | |
| Apr 16, 2023 at 13:55 | comment | added | LSpice | @HAHelfgott, re, which maps? The map $U^- \times T \times U^+ \to G$ is just the restriction of the multiplication $G \times G \times G \to G$, which is certainly rational. Groups possess maximal tori by Grothendieck's theorem, geometric unipotent radicals are rational (and split) because finite fields are perfect, and groups over finite fields have rational Borels essentially because Galois cohomology vanishes by Lang–Steinberg. | |
| Apr 16, 2023 at 7:30 | comment | added | H A Helfgott | I take it is easy to see that all of these maps are defined over the ground field? | |
| Apr 16, 2023 at 1:37 | comment | added | LSpice | If you're willing for the bound to depend also on $d_\text s$ and $r$, then you can probably do even better than $q^{d - r}(q - 1)^r$ by observing that (whether or not $G$ is reductive) $G/B^+$ is a projective variety of dimension $\frac1 2 d_\text s$; but I don't know much about point counts on projective varieties. | |
| Apr 16, 2023 at 0:34 | history | answered | LSpice | CC BY-SA 4.0 |