Timeline for Are centrally extended p-adic groups defined over F_1?
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| May 14, 2010 at 10:14 | comment | added | JBorger | You can see the final section of arxiv:0906.3146, but unfortunately there's not much there. I think we should talk. | |
| May 14, 2010 at 8:57 | comment | added | André Henriques | Great! What is the p-typical F_1? Where can I read about it? | |
| May 14, 2010 at 5:11 | comment | added | JBorger | I was too cautious. "$G(\mathbf{Q}_p)$" is definitely a group-ind-scheme defined over the $p$-typical $\mathbf{F}_1$ as long as $G$ is an affine group scheme. (It would take a bit of space to explain this.) You should try to make the central extension then! | |
| May 14, 2010 at 5:03 | comment | added | JBorger | I haven't ever really thought about the passage to the group-ind-scheme $G(\mathbf{Q}_p)$, so I don't know whether it's possible to descend $G(\mathbf{Q}_p)$ similarly, but it seems pretty reasonable. I also don't know anything about the extension, but it seems like a reasonable thing to consider. Also, I have heard that William Haboush has a manuscript on $G(\mathbf{Q}_p)$-like group-ind-schemes, so you might consult him. | |
| May 14, 2010 at 4:59 | comment | added | JBorger | There might be a bit more to the original question. From the lambda/Witt point of view on $\mathbf{F}_1$, "$G(\mathbf{Z}_p)$" (or rather the Greenberg functor applied to $G$) is not just an a group scheme over $\mathbf{F}_p$, it also the base change to $\mathbf{F}_p$ of a group scheme over the "$p$-typical $\mathbf{F}_1$". In this context, this just means that "$G(\mathbf{Z}_p)$", if we view it as a flat group scheme over $\mathbf{Z}_p$ (rather than just looking at its special fiber) has a lift of the Frobenius map. (continued) | |
| May 14, 2010 at 4:03 | history | edited | Torsten Ekedahl | CC BY-SA 2.5 |
Added concrete example
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| May 14, 2010 at 1:20 | comment | added | George McNinch | @Henriques: the algebraic group in that case represents the functor which assigns to a $\mathbf{Z}/p$ algebra $B$ the unit group $W_2(B)^\times$ of the length 2 Witt vectors $W_2(B)$. In maybe more down-to-earth language, take an algebraic closure $k$ of $\mathbf{Z}/p$. The unit group of $W_2(k)$ is an algebraic group which is defined over $\mathbf{Z}/p$ and is the required group. | |
| May 13, 2010 at 19:55 | vote | accept | André Henriques | ||
| May 13, 2010 at 19:34 | comment | added | André Henriques | Thank you Torsten for your very nice answer. You say that G(Z/p^n) are the Z/p points of a Z/p-algebraic group. Could you maybe illustrate that claim in the simple case G=G_m and n=2. What is the algebraic group in that case? | |
| May 13, 2010 at 19:25 | history | answered | Torsten Ekedahl | CC BY-SA 2.5 |