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

Abstract

In this paper we prove local-global principles for the existence of an embedding (E,σ) (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local-global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S-arithmetic groups in all absolutely simple algebraic groups of type different from D4 which was initiated in our paper [23]. More precisely, we prove that in a group of type Dn, 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 + 7, with n ≥ 1. The appendix contains a Galois-cohomological interpretation of our embedding theorems. © Swiss Mathematical Society.