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

Abstract

Let $G_1$ and $G_2$ be absolutely almost simple algebraic groups of types $B_{\ell }$ and $C_{\ell }$, 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.