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Tr. Mat. Inst. Steklova, 2016, Volume 292, Pages 224–254 (Mi tm3689)  

On the congruence kernel for simple algebraic groups

Gopal Prasada, Andrei S. Rapinchukb

a Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA
b Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA

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.

Funding Agency Grant Number
National Science Foundation DMS-1401380
DMS-1301800
Both authors were supported by the NSF (grants DMS-1401380 and DMS-1301800).


DOI: https://doi.org/10.1134/S0371968516010143

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English version:
Proceedings of the Steklov Institute of Mathematics, 2016, 292, 216–246

Bibliographic databases:

Document Type: Article
UDC: 512.74
Received: January 11, 2015
Language: English

Citation: Gopal Prasad, Andrei S. Rapinchuk, “On the congruence kernel for simple algebraic groups”, Algebra, geometry, and number theory, Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday, Tr. Mat. Inst. Steklova, 292, MAIK Nauka/Interperiodica, Moscow, 2016, 224–254; Proc. Steklov Inst. Math., 292 (2016), 216–246

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\paper On the congruence kernel for simple algebraic groups
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\bookinfo Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday
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\pages 224--254
\publ MAIK Nauka/Interperiodica
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