Timeline for 'Adelic torus' not arising from a rational torus
Current License: CC BY-SA 3.0
20 events
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| Nov 12, 2016 at 19:26 | comment | added | nfdc23 | Existence of $(G_R)_{\gamma}$ is easy: inside $G_R \times G_R$, intersect the diagonal copy of $G_R$ with the graph of the $\gamma$-conjugation map. That's all! (Of course, there is content in the fact that when $R$ is a field and $\gamma$ is a semisimple rational point then this schematic centralizer is actually smooth with the "expected" Lie algebra.) To do anything one needs properties of that schematic centralizer (e.g., $R$-smoothness? An $R$-torus?), and that requires knowing more about your situation. There is no notion of semisimple $R$-point, so no off-the-shelf result in SGA3 . | |
| Nov 12, 2016 at 17:11 | comment | added | Tian An | You're right, I think this is what i want. Do you have a reference for this, say, SGA3? Long story short, the application is to be able to apply a kind of Poisson formula to the geometric side of a trace formula, which is indexed by rational conj classes $\gamma$, and whose terms involve volumes of tori defined as centralizers of $\gamma$. But for a Poisson formula one has to be to interpret these terms as functions on $\mathbb A_F$, not just $F$. | |
| Nov 12, 2016 at 13:43 | comment | added | nfdc23 | As I said above, for any $F$-algebra $R$ (not just $\mathbf{A}_F$) and $\gamma \in G(R)$ we can form the centralizer scheme $(G_R)_{\gamma}$ as a closed $R$-subgroup scheme of $G_R$. In what sense is this not what you want for "extending" the usual notion of $G_{\gamma}$ for $\gamma \in G(F)$ to the case of points of $G$ valued in more general $F$-algebras? It remains unclear precisely what is the motivation for the original question (in terms of what you had in mind to do with an answer to it). | |
| Nov 11, 2016 at 20:23 | comment | added | Tian An | (MO is asking to avoid extended discussions, so if you wrote a reply this might help.) In any case, the 'generalization' i'm looking for is interpreting $G_\gamma$ as a function of $\gamma \in G(F)$, and extending it to $\gamma\in G(\mathbb A_F)$. I'm not sure I understand your objection to (1), but the conjugation should be at each place $G(F_v)$ | |
| Nov 11, 2016 at 19:47 | comment | added | nfdc23 | Since the hypotheses in your (1) are insensitive to applying any $G(\mathbf{A}_F)$-conjugation, it seems highly unlikely that anything like those hypotheses could produce the group of adelic points of an $F$-torus; you need more global data than what is written in your condition (1). As for (2), many people use centralizer schemes in reductive groups over rings, but since it isn't clear what you want to do with a potential "generalization" of adelic points of an $F$-torus it is difficult to know what to suggest about this. | |
| Nov 11, 2016 at 18:12 | comment | added | Tian An | @nfdc23 Thanks for the comment. Right, I don't expect to always get an $F$-torus, my questions were (1) if my data would give an $F$-torus given the condition above (2) if this more general definition has been studied in the literature. | |
| Nov 11, 2016 at 14:22 | comment | added | nfdc23 | One can define the centralizer scheme $(G_R)_{\gamma}$ as a closed $R$-subgroup scheme of $G_R$ for any $F$-algebra $R$ (e.g., $R = \mathbf{A}_F$) and $\gamma \in G(R)$, and ask if this is an $R$-torus (in the sense of SGA3). There is certainly no way to make an $F$-torus inside $G$ from an adelic point, but presumably you are well aware of that; the only reasonable expectation is to make a torus over the $F$-algebra for which the "point" $\gamma$ is given (e.g., over $\mathbf{A}_F$). But your hypothesis on the $\gamma_v$'s makes no contact with any integral structures, so it is very unlikely. | |
| Nov 11, 2016 at 13:43 | comment | added | Tian An | Oh, yes, I did mean to take a restricted direct product. The motivation, roughly speaking, is I would like to define a torus $T=G_\gamma$ using the data of the group $G$ and an adelic $\gamma\in G_{\mathbb A}$, in a way that is compatible with the adelic torus obtained from $T(F)$, i.e., a rational $\gamma$. | |
| Nov 11, 2016 at 13:37 | comment | added | nfdc23 | The space $T(\mathbf{A}_F)$ is not $\prod_v T(F_v)$; it is a tiny subgroup of the latter; rather, $T(\mathbf{A}_F)$ coincides with a "restricted product" defined by an affine finite type $O_{F,S}$-scheme $\mathscr{T}$ with generic fiber $T$ (all choices of which give the same notion of restricted product). So what exactly are you asking about with $\prod T_v(F_v)$ for unrelated $T_v$'s over $F_v$'s for all $v$? There's no meaningful $S$-integral "glue", so there is no reasonable notion of "restricted product", hence not inside $G(\mathbf{A}_F)$. What is the actual motivating situation? | |
| Nov 11, 2016 at 12:51 | comment | added | Tian An | Ok, I've edited that out. Hope it is clearer now. | |
| Nov 11, 2016 at 12:50 | history | edited | Tian An | CC BY-SA 3.0 |
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| Nov 11, 2016 at 12:21 | review | Close votes | |||
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| Nov 11, 2016 at 12:11 | comment | added | Mikhail Borovoi | I mean your English is not quite clear. | |
| Nov 11, 2016 at 12:10 | comment | added | Tian An | By definition of $\gamma$, since the centralizer of a strongly regular semisimple element is a torus. | |
| Nov 11, 2016 at 12:09 | history | edited | Tian An | CC BY-SA 3.0 |
edited for clarity
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| Nov 11, 2016 at 12:09 | comment | added | Mikhail Borovoi | It is not clear what your are asking. What is the question? What does it mean: "By definition the centralizer $G_\gamma$ is defines an $F$-torus $T$" ? | |
| Nov 11, 2016 at 12:08 | history | edited | Tian An | CC BY-SA 3.0 |
edited for clarity
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| Nov 11, 2016 at 10:57 | history | edited | Tian An | CC BY-SA 3.0 |
added 171 characters in body
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| Nov 11, 2016 at 10:46 | history | edited | Tian An |
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| Nov 11, 2016 at 10:37 | history | asked | Tian An | CC BY-SA 3.0 |