Timeline for Existence of a model over S-integers
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 4, 2017 at 13:50 | answer | added | nfdc23 | timeline score: 5 | |
| Dec 4, 2017 at 6:47 | comment | added | David Loeffler | I think this question is quite sensitive to the exact meaning of the word "model". You want to find some scheme $\mathcal{G} / \mathcal{O}_{F, S}$ such that $\mathcal{G} \times F = G$. But do you require $\mathcal{G}$ to be a reductive group scheme, an arbitrary group scheme, or a totally arbitrary scheme? | |
| Dec 3, 2017 at 21:18 | comment | added | Daniel Loughran | @CoffeeBliss: Your changes have not solved the problem; the answer to your question is still no, as the example of user94041 still stands. | |
| Dec 3, 2017 at 21:14 | comment | added | Not a grad student | @PeterMcNamara For almost all places $G \times_F F_v$ is quasisplit and is split over an unramified extension. But I am wondering what happens if $S$ is smaller than that set. | |
| Dec 3, 2017 at 21:13 | comment | added | Not a grad student | @user94041 I edited my question to include conditions on $S$. | |
| Dec 3, 2017 at 20:00 | history | edited | Not a grad student | CC BY-SA 3.0 |
added 34 characters in body
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| Dec 3, 2017 at 19:35 | comment | added | Peter McNamara | If you allow yourself the possibility of adding a finite number of places to S, then the answer becomes yes. | |
| Dec 3, 2017 at 18:01 | comment | added | user94041 | What if I take $\mathrm G$ to be the norm 1 units of a quadratic extension $E / F$? It seems that I can write down the coordinate ring of $\mathrm G$ over the ring of integers of $F$, so I can take $S$ to be empty. But at places $v$ where $E/F$ is ramified, you will need a ramified extension to split the torus. | |
| Dec 3, 2017 at 17:25 | comment | added | YCor | "Suppose there is a set $S$ of places for which $G$ has a model over the ring of $S$-integers in $F$" is an empty condition. I reformulated accordingly. | |
| Dec 3, 2017 at 17:24 | history | edited | YCor | CC BY-SA 3.0 |
clarified
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| Dec 3, 2017 at 17:19 | history | asked | Not a grad student | CC BY-SA 3.0 |