Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the congruence kernel for simple algebraic groups   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.

7
The categories of this paper are: hep-th
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
26
The categories of this paper are: bayes-an
1
The categories of this paper are: cs
28
The categories of this paper are: alg-geom
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
30
The categories of this paper are: q-alg
35
The categories of this paper are: mtrl-th
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
34
The categories of this paper are: chem-ph
15
The categories of this paper are: nucl-ex
33
The categories of this paper are: plasm-ph