On the congruence kernel for simple algebraic groups

TL;DR

This paper investigates the structure of the congruence kernel in algebraic groups over global fields, providing conditions for triviality and centrality, and revealing its generation properties in specific cases.

Contribution

It offers new criteria for the triviality and centrality of the congruence kernel, and describes its generation in the case of K-isotropic groups over number fields.

Findings

C^(S)(G) is trivial if S contains a generalized arithmetic progression.

Provides a criterion for the centrality of C^(S)(G) based on commuting lifts.

C^(S)(G) is almost generated by a single element when G is K-isotropic over a number field.

Abstract

This paper contains several results about the structure of the congruence kernel C^(S)(G) of an absolutely almost simple simply connected algebraic group G over a global field K with respect to a set of places S of K. In particular, we show that C^(S)(G) is always trivial if S contains a generalized arithmetic progression. We also give a criterion for the centrality of C^(S)(G) in the general situation in terms of the existence of commuting lifts of the groups G(K_v) for v \notin S in the S-arithmetic completion \widehat{G}^(S). This result enables one to give simple proofs of the centrality in a number of cases. Finally, we show that if K is a number field and $G$ is K-isotropic then C^(S)(G) as a normal subgroup of \widehat{G}^(S) is almost generated by a single element.