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

TL;DR

This paper fully characterizes the structure of the congruence kernel of an arithmetic lattice in a rank one algebraic group over a local field, revealing it depends only on the characteristic of the base field.

Contribution

It provides a complete description of the congruence kernel for such lattices, showing it is a free profinite product with a factor being a free profinite group, and depends solely on the characteristic of the field.

Findings

The congruence kernel is a free profinite product including a free profinite group on countably many generators.

The structure of the congruence kernel depends only on the characteristic of the base field.

The results unify and extend known cases of the congruence subgroup problem.

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 \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 \Gamma be an arithmetic lattice in G and let C=C(\Gamma) 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}_{\omega}, 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=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 \Gamma on the Bruhat-Tits tree associated with G.