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

TL;DR

This paper establishes local-global principles for embedding fields with involution into simple algebras with involution over global fields, impacting the study of classical groups and hyperbolic spaces.

Contribution

It proves new local-global principles for embeddings with involution, leading to results on group isomorphisms and properties of hyperbolic spaces.

Findings

Weakly commensurable S-arithmetic subgroups are actually commensurable.

K-forms with same maximal tori are K-isomorphic.

Results apply to hyperbolic spaces of specific dimensions.

Abstract

In this paper we prove local-global principles for embedding of fields with involution into central simple algebras with involution over a global field. These should be of interest in study of classical groups over global fields. We deduce from our results that in a group of type D_n, n>4 even, two weakly commensurable Zariski-dense S-arithmetic subgroups are actually commensurable. A consequence of this result is that given an absolutely simple algebraic K-group G of type D_n, n>4 even, K a number field, any K-form G' of G having the same set of isomorphism classes of maximal K-tori as G, is necessarily K-isomorphic to G. These results lead to results about isolength and isospectral compact hyperbolic spaces of dimension 2n-1 with n even.