Timeline for Orbital integral in Cluckers and Denef
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
|
|
| Jun 7, 2020 at 19:35 | vote | accept | Tian An | ||
| Jun 6, 2020 at 16:58 | answer | added | JGordon | timeline score: 3 | |
| Apr 10, 2020 at 14:44 | comment | added | LSpice | Thanks, but I think that there's space for a much better answer. I e-mailed @JGordon (second author of the CGH paper above), who's not an MO regular but of course knows the ins and outs of this. | |
| Apr 10, 2020 at 4:03 | comment | added | Tian An | Fair enough. Unfortunately I was hoping the theorem said something about more general test functions. In any case, if you'd like to copy what you've written as an answer I'd be happy to accept it @LSpice :) | |
| Apr 10, 2020 at 3:57 | comment | added | LSpice | Yes. I agree it's rather narrow, but it's what you need for the fundamental lemma, so it receives quite a lot of attention. I'm not familiar enough with both papers to say, but the CGH paper may give the details of how to make this connection more general. | |
| Apr 10, 2020 at 3:56 | comment | added | LSpice | You may also find helpful Section 4 of the later paper by one of the same authors, Cluckers, Gordon, and Halupczok - Uniform analysis on local fields and applications to orbital integrals (MSN). | |
| Apr 10, 2020 at 3:56 | comment | added | Tian An | Thanks, @LSpice! By $[\mathscr K]$ do you mean the characteristic function of $\mathscr K$? That would be a rather narrow class of orbital integrals.. | |
| Apr 10, 2020 at 3:54 | history | edited | LSpice | CC BY-SA 4.0 |
Minor TeX fixes
|
| Apr 10, 2020 at 3:45 | comment | added | LSpice | Actually, wait, I think you need only take $f = 1$, and $g_i$s to be obtained from the $f_i$s of Theorem 1.1 by taking appropriate scales and shifts so that $g_i^{-1}(\mathcal O_K)$ detects $\operatorname{ord}(f_i(x))$ and $g_i^{-1}(\mathcal O_K)$ detects $\overline{\operatorname{ac}}(f_i(x))$. (More general $f$, and the parameter $s$, are probably for the application to Igusa's work, not for orbital integrals in the sense with which you and I are familiar.) | |
| Apr 10, 2020 at 3:41 | comment | added | LSpice | I think that these are meant to capture, not fully general orbital integrals, but something like your $O_\gamma([\mathscr K])$, where $\mathscr K$ is a hyperspecial parahoric subgroup. Then one may take, in the case of $\operatorname{GL}_n$, $g_i$ to be the various co-ordinate functions, and $f$ to run over the various $f_i$'s of Theorem 1.1 to recover $O_\gamma([\mathscr K])$ from the various $I_{K, x}(s)$. (Incidentally, I think you want $G(F^{\text a})$ acting on $X(F^{\text a})$, not $\mathbb C$-rational points.) | |
| Apr 10, 2020 at 3:29 | history | asked | Tian An | CC BY-SA 4.0 |