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

### Gopal Prasad

University of Michigan, Ann Arbor, United States### Andrei S. Rapinchuk

University of Virginia, Charlottesville, USA

## 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 $D_{4}$ 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 $n≥1$. The appendix contains a Galois-cohomological interpretation of our embedding theorems.

## Cite this article

Gopal Prasad, Andrei S. Rapinchuk, Local–global principles for embedding of fields with involution into simple algebras with involution. Comment. Math. Helv. 85 (2010), no. 3, pp. 583–645

DOI 10.4171/CMH/206