The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field

Mason, A., Premet, A., Sury, B. and Zalesskii, P.A. (2008) The congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field. Journal für die reine und angewandte Mathematik (Crelles Journal), 623, pp. 43-72. (doi: 10.1515/CRELLE.2008.072)

Abstract

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 G be a connected, simply-connected algebraic group over k, which is absolutely almost simple of k(v)-rank 1. Let G = G(k(v)). Let Gamma be an arithmetic lattice in G and let C = C(Gamma) be its congruence kernel. Lubotzky has shown that C is infinite, conforming an earlier conjecture of Serre. Here we provide complete solution of the congruence subgroup problem for Gamma by determining the structure of C. It is shown that C is a free profinite product, one of whose factors is (F-omega) over cap, 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 Gamma = SL2(O(S)), where 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 Gamma on the Bruhat-Tits tree associated with G.

• Item Type: Articles
• Status: Published
• Refereed: Yes
• Glasgow Author(s) Enlighten ID: Mason, Dr Alexander
• Authors: Mason, A., Premet, A., Sury, B., and Zalesskii, P.A.
• College/School: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
• Journal Name: Journal für die reine und angewandte Mathematik (Crelles Journal)
• ISSN: 0075-4102