Timeline for Lattices in $p$-adic groups
Current License: CC BY-SA 4.0
13 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 14 at 2:56 | vote | accept | Jun Yang | ||
| Dec 18, 2022 at 6:20 | comment | added | Asaf | One recent reference is Morris' book about arithmetic groups, this construction is from section 6.3 (also 6.7,6.8). It might be in Borel's book as well, but I am not sure and I don't have the book with me. | |
| Dec 18, 2022 at 3:53 | vote | accept | Jun Yang | ||
| Dec 19, 2022 at 15:39 | |||||
| Dec 18, 2022 at 3:53 | comment | added | Jun Yang | @Asaf Thank you so much! BTW, is there any reference for this construction? It seems beyond Margulis’ book. | |
| Dec 17, 2022 at 20:29 | comment | added | Asaf | @JunYang , yes, it comes from a an arithmetic construction. The full examples starts as a suitable subgroup fo the $\mathbb{Q}$-grp $GL_{2d}(\mathbb{C})$, with $S={\infty, p}$. The thing one needs to observe is that at the infinity place, $G_{\mathbb{R}}\simeq SU_{d}(\mathbb{R})$ is compact, hence the projection of the lattice to the other factor is still a lattice. | |
| Dec 17, 2022 at 20:24 | comment | added | Asaf | @YCor , thanks, I think I should have restricted $SL_{d}(\mathbb{Z}[1/(p_{1}\cdot p_{2})])$ to these lattices with $g^{t}\cdot g = I$ and take it in the product of $SO_{d}(\mathbb{Q}_{p_{1}})\times SO_{d}(\mathbb{Q}_{p_2})$, but for now I deleted it. | |
| Dec 17, 2022 at 20:21 | history | edited | Asaf | CC BY-SA 4.0 |
deleted a wrong example
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| Dec 17, 2022 at 19:51 | comment | added | Jun Yang | @Asaf Thank you for the example in p-adic $SL_d$. Is that arithmetic? Or what is the definition of arithmeticity in p-adic groups without a set S containing any infinite primes? | |
| Dec 17, 2022 at 16:48 | comment | added | YCor | But $\mathrm{SL}_d(\mathbf{Z}[1/pq])$ is not a lattice in $\mathrm{SL}_d(\mathbf{Q}_p)\times\mathrm{SL}_d(\mathbf{Q}_q)$ (it is dense!). It is a lattice in $\mathrm{SL}_d(\mathbf{R})\times\mathrm{SL}_d(\mathbf{Q}_p)\times\mathrm{SL}_d(\mathbf{Q}_q)$. | |
| Dec 17, 2022 at 15:40 | comment | added | Asaf | Forgot $SL_{d}$... | |
| Dec 17, 2022 at 4:29 | history | edited | Asaf | CC BY-SA 4.0 |
added 17 characters in body
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| Dec 16, 2022 at 22:31 | comment | added | YCor | I don't understand your last assertion. | |
| Dec 16, 2022 at 20:06 | history | answered | Asaf | CC BY-SA 4.0 |