RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PERSONAL OFFICE
 General information Latest issue Forthcoming papers Archive Impact factor Guidelines for authors License agreement Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Trudy MIAN: Year: Volume: Issue: Page: Find

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

On the congruence kernel for simple algebraic groups

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-1401380DMS-1301800 Both authors were supported by the NSF (grants DMS-1401380 and DMS-1301800).

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

Full text: PDF file (445 kB)
References: PDF file   HTML file

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

Bibliographic databases:

Document Type: Article
UDC: 512.74
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

Citation in format AMSBIB
\Bibitem{PraRap16} \by Gopal~Prasad, Andrei~S.~Rapinchuk \paper On the congruence kernel for simple algebraic groups \inbook Algebra, geometry, and number theory \bookinfo Collected papers. Dedicated to Academician Vladimir Petrovich Platonov on the occasion of his 75th birthday \serial Tr. Mat. Inst. Steklova \yr 2016 \vol 292 \pages 224--254 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3689} \crossref{https://doi.org/10.1134/S0371968516010143} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3628463} \elib{http://elibrary.ru/item.asp?id=25772722} \transl \jour Proc. Steklov Inst. Math. \yr 2016 \vol 292 \pages 216--246 \crossref{https://doi.org/10.1134/S0081543816010144} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000376271200014} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84971268646}