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 $(E,\sigma )\hookrightarrow (A,\tau )$ of a given global field $E$ endowed with an involutive automorphism $\sigma$ into a simple algebra $A$ given with an involution $\tau$ in all situations except where $A$ is a matrix algebra of even degree over a quaternion division algebra and $\tau$ 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 $(A,\tau )$ 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 $D_4$ which was initiated in our paper [23]. 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 $4n+7$, with $n\ge 1$. The appendix contains a Galois-cohomological interpretation of our embedding theorems.