The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field
| dc.creator | Mason, A. W. | |
| dc.creator | Premet, A. | |
| dc.creator | Sury, B. | |
| dc.creator | Zalesskii, P. A. | |
| dc.date | 2007-10-22 | |
| dc.date.accessioned | 2026-07-07T08:37:43Z | |
| dc.date.available | 2026-07-07T08:37:43Z | |
| dc.description | Let k be a global field and let k_v be the completion of k with respect to v, a non-archimedean place of k. Let \mathbf{G} be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k_v-rank 1. Let G=\mathbf{G}(k_v). Let Γbe an arithmetic lattice in G and let C=C(Γ) be its congruence kernel. Lubotzky has shown that C is infinite, confirming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for \Gamm$ by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is \hat{F}_ω, the free profinite group on countably many generators. The most surprising conclusion from our results is that the structure of C depends only on the characteristic of k. The structure of C is already known for a number of special cases. Perhaps the most important of these is the (non-uniform) example Γ=SL_2(\mathcal{O}(S)), where \mathcal{O}(S) is the ring of S-integers in k, with S=\{v\}, which plays a central role in the theory of Drinfeld modules. The proof makes use of a decomposition theorem of Lubotzky, arising from the action of Γon the Bruhat-Tits tree associated with G. | |
| dc.description | 27 pages, 5 figures, to appear in J. Reine Angew. Math | |
| dc.identifier | https://arxiv.org/abs/0710.4009 | |
| dc.identifier | http://arxiv.org/abs/0710.4009 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/140498 | |
| dc.subject | Group Theory | |
| dc.subject | Number Theory | |
| dc.title | The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field | |
| dc.type | text |