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

TL;DR

This paper characterizes when certain algebraic groups of types B and C over number fields have equivalent maximal tori, leading to criteria for weak commensurability of their S-arithmetic subgroups.

Contribution

It provides necessary and sufficient conditions for weak commensurability of S-arithmetic subgroups in types B and C algebraic groups, based on their maximal tori classifications.

Findings

Criteria for isomorphism or isogeny classes of maximal K-tori

Conditions for weak commensurability of S-arithmetic subgroups

Application to algebraic groups of ranks ≥ 3

Abstract

Let G and G' be absolutely almost simple algebraic groups of types B and C respectively, of rank at least 3, and defined over a number field K. We determine when G and G' 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 and G' to be weakly commensurable.