Weakly commensurable arithmetic groups and isospectral locally symmetric spaces

Local–global principles for embedding of fields with involution into simple algebras with involution

“…We showed in [23], for a general absolutely almost simple algebraic F -group G, that if i is a Zariski-dense .G i ; K i ; S i /-arithmetic subgroup of G.F / for i D 1; 2, then the weak commensurability of 1 and 2 implies that K 1 D K 2 and S 1 D S 2 (Theorem 3 of [23]), and then their commensurability up to an F -automorphism of x G is equivalent to the assertion that G 1 ' G 2 over K (Proposition 2.5 of [23]). Furthermore, we showed that the latter follows from weak commensurability of 1 and 2 if G is of type different from A n , D n , and E 6 .…”

“…More precisely, we prove that in a group of type D n , n even > 4, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n C 7, with n > 1. The appendix contains a Galois-cohomological interpretation of our embedding theorems.…”

“…In this section, we will show how our previous results (particularly, Theorem 8.1) can be used to complete the analysis of weakly commensurable arithmetic subgroups in the case that was left open in the original version of [23], viz. where the ambient algebraic groups are of type D 2r , with r > 3 (for obvious reasons, type D 4 requires a special treatment, we hope to study groups of this type later).…”

“…where the ambient algebraic groups are of type D 2r , with r > 3 (for obvious reasons, type D 4 requires a special treatment, we hope to study groups of this type later). We first recall the notion of weak commensurability introduced in [23]. Let G 1 and G 2 be two connected semi-simple algebraic groups defined over a field F of characteristic zero.…”

“…So, by replacing the given groups G i 's with isogenous ones and enlarging the field F if necessary, we reduce our analysis of weakly commensurable subgroups to the situation where G 1 D G 2 D G, and moreover, G is a F -form of SO n , hence F -isomorphic to SU.A; / for some central simple n 2 -dimensional F -algebra A with an orthogonal involution . One of the central issues in [23] was to determine when weak commensurability of S-arithmetic subgroups implies their commensurability, which in turn led to some interesting results about length-commensurable and isospectral locally symmetric spaces (see [24] for a nontechnical exposition of these results). We used the following definition of S -arithmeticity.…”