Weakly commensurable \(S\)-arithmetic subgroups in almost simple algebraic groups of types B and C. (English) Zbl 1285.20045

Garibaldi, Skip; Rapinchuk, Andrei

Algebra Number Theory 7, No. 5, 1147-1178 (2013).

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 \(K\). The authors determine when \(G_1\) and \(G_2\) have the same isomorphism or isogeny classes of maximal \(K\)-tori. This leads to necessary and sufficient conditions for two Zariski-dense \(S\)-arithmetic subgroups of \(G_1\) and \(G_2\) to be weakly commensurable.

Reviewer: L. N. Vaserstein (University Park)

MSC:

20G30	Linear algebraic groups over global fields and their integers
11E57	Classical groups
14L35	Classical groups (algebro-geometric aspects)
20G15	Linear algebraic groups over arbitrary fields

Keywords:

simple algebraic groups; isogenous maximal tori; weak commensurability; isomorphic maximal tori; Zariski-dense \(S\)-arithmetic subgroups