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This article is cited in 5 scientific papers (total in 5 papers)
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.
Received: January 11, 2015
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, Trudy Mat. Inst. Steklova, 292, MAIK Nauka/Interperiodica, Moscow, 2016, 224–254; Proc. Steklov Inst. Math., 292 (2016), 216–246
Linking options:
https://www.mathnet.ru/eng/tm3689https://doi.org/10.1134/S0371968516010143 https://www.mathnet.ru/eng/tm/v292/p224
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Abstract page: | 168 | Full-text PDF : | 49 | References: | 79 | First page: | 3 |
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