Weakly commensurable S-arithmetic subgroups in almost simple algebraic groups of types B and C

Garibaldi, Skip; Rapinchuk, Andrei

doi:10.2140/ant.2013.7.1147

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Weakly commensurable $S$-arithmetic subgroups in almost simple algebraic groups of types $\mathsf{B}$ and $\mathsf{C}$

Skip Garibaldi and Andrei Rapinchuk

Vol. 7 (2013), No. 5, 1147–1178

DOI: 10.2140/ant.2013.7.1147

Abstract

Let $G_{1}$ and $G_{2}$ be absolutely almost simple algebraic groups of types $B_{ℓ}$ and $C_{ℓ}$ , respectively, defined over a number field $K$ . We determine when $G_{1}$ and $G_{2}$ have the same isomorphism or isogeny classes of maximal $K$ -tori. This leads to the necessary and sufficient conditions for two Zariski-dense $S$ -arithmetic subgroups of $G_{1}$ and $G_{2}$ to be weakly commensurable.

Keywords

isogenous maximal tori, weak commensurability, isomorphic maximal tori

Mathematical Subject Classification 2010

Primary: 20G15

Secondary: 11E57, 14L35, 20G30

Milestones

Received: 20 January 2012

Revised: 29 April 2012

Accepted: 7 June 2012

Published: 6 September 2013

Authors

Skip Garibaldi
Institute for Pure and Applied Mathematics 460 Portola Plaza Box 957121 Los Angeles, CA 90095-7121 United States
http://www.mathcs.emory.edu/~skip/
Andrei Rapinchuk
Department of Mathematics University of Virginia Charlottesville, VA 22904 United States
http://www.math.virginia.edu/Faculty/Rapinchuk/

Vol. 7, No. 5, 2013